CoCalc -- S2.qti

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²⁰⁸⁶⁵ views
<?xml version='1.0' encoding='UTF-8'?>
<questestinterop xmlns="http://www.imsglobal.org/xsd/ims_qtiasiv1p2" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.imsglobal.org/xsd/ims_qtiasiv1p2 http://www.imsglobal.org/xsd/ims_qtiasiv1p2p1.xsd">
  <objectbank ident="S2">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- S2</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="S2-2012" title="S2 | Linear ODE system with elimination | ver. 2012"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= 4" alt="2 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= 4" title="2 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= 4" data-latex="2 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= 4"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y} + {y'} = -12 \, {x} \hspace{2em}x(0)= 1" alt="-3 \, {y} + {y'} = -12 \, {x} \hspace{2em}x(0)= 1" title="-3 \, {y} + {y'} = -12 \, {x} \hspace{2em}x(0)= 1" data-latex="-3 \, {y} + {y'} = -12 \, {x} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%20-3%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%204" alt="2 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= 4" title="2 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= 4" data-latex="2 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= 4"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%20-12%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-3 \, {y} + {y'} = -12 \, {x} \hspace{2em}x(0)= 1" title="-3 \, {y} + {y'} = -12 \, {x} \hspace{2em}x(0)= 1" data-latex="-3 \, {y} + {y'} = -12 \, {x} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}" alt="x= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" alt="y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-3%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6373" title="S2 | Linear ODE system with elimination | ver. 6373"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} = {x'} + {x} \hspace{2em}x(0)= 0" alt="-{y} = {x'} + {x} \hspace{2em}x(0)= 0" title="-{y} = {x'} + {x} \hspace{2em}x(0)= 0" data-latex="-{y} = {x'} + {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 4 \, {x} - 6 \, {y} - {y'} \hspace{2em}x(0)= -3" alt="0 = 4 \, {x} - 6 \, {y} - {y'} \hspace{2em}x(0)= -3" title="0 = 4 \, {x} - 6 \, {y} - {y'} \hspace{2em}x(0)= -3" data-latex="0 = 4 \, {x} - 6 \, {y} - {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By%7D%20=%20%7Bx'%7D%20+%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y} = {x'} + {x} \hspace{2em}x(0)= 0" title="-{y} = {x'} + {x} \hspace{2em}x(0)= 0" data-latex="-{y} = {x'} + {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%204%20%5C,%20%7Bx%7D%20-%206%20%5C,%20%7By%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = 4 \, {x} - 6 \, {y} - {y'} \hspace{2em}x(0)= -3" title="0 = 4 \, {x} - 6 \, {y} - {y'} \hspace{2em}x(0)= -3" data-latex="0 = 4 \, {x} - 6 \, {y} - {y'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" alt="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5231" title="S2 | Linear ODE system with elimination | ver. 5231"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 1" alt="4 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 1" title="4 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 1" data-latex="4 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y'} - 2 \, {x} - 9 \, {y} \hspace{2em}x(0)= 1" alt="0 = -{y'} - 2 \, {x} - 9 \, {y} \hspace{2em}x(0)= 1" title="0 = -{y'} - 2 \, {x} - 9 \, {y} \hspace{2em}x(0)= 1" data-latex="0 = -{y'} - 2 \, {x} - 9 \, {y} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="4 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 1" title="4 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 1" data-latex="4 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By'%7D%20-%202%20%5C,%20%7Bx%7D%20-%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = -{y'} - 2 \, {x} - 9 \, {y} \hspace{2em}x(0)= 1" title="0 = -{y'} - 2 \, {x} - 9 \, {y} \hspace{2em}x(0)= 1" data-latex="0 = -{y'} - 2 \, {x} - 9 \, {y} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" alt="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4709" title="S2 | Linear ODE system with elimination | ver. 4709"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -5" alt="-4 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -5" title="-4 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -5" data-latex="-4 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} - {y} = -{y'} \hspace{2em}x(0)= 0" alt="-{x} - {y} = -{y'} \hspace{2em}x(0)= 0" title="-{x} - {y} = -{y'} \hspace{2em}x(0)= 0" data-latex="-{x} - {y} = -{y'} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20=%20-2%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-4 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -5" title="-4 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -5" data-latex="-4 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20-%20%7By%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x} - {y} = -{y'} \hspace{2em}x(0)= 0" title="-{x} - {y} = -{y'} \hspace{2em}x(0)= 0" data-latex="-{x} - {y} = -{y'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" alt="x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="y= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1552" title="S2 | Linear ODE system with elimination | ver. 1552"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 3" alt="-{x'} = -4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 3" title="-{x'} = -4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 3" data-latex="-{x'} = -4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 0" alt="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 0" title="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 0" data-latex="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-4%20%5C,%20%7By%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-{x'} = -4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 3" title="-{x'} = -4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 3" data-latex="-{x'} = -4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-9%20%5C,%20%7By%7D%20+%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 0" title="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 0" data-latex="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" alt="y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6834" title="S2 | Linear ODE system with elimination | ver. 6834"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 3" alt="2 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 3" title="2 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 3" data-latex="2 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 6 \, {y} = 2 \, {x} \hspace{2em}x(0)= 3" alt="{y'} + 6 \, {y} = 2 \, {x} \hspace{2em}x(0)= 3" title="{y'} + 6 \, {y} = 2 \, {x} \hspace{2em}x(0)= 3" data-latex="{y'} + 6 \, {y} = 2 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20+%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="2 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 3" title="2 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 3" data-latex="2 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%206%20%5C,%20%7By%7D%20=%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{y'} + 6 \, {y} = 2 \, {x} \hspace{2em}x(0)= 3" title="{y'} + 6 \, {y} = 2 \, {x} \hspace{2em}x(0)= 3" data-latex="{y'} + 6 \, {y} = 2 \, {x} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" alt="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(-2 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(-2 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-2 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6599" title="S2 | Linear ODE system with elimination | ver. 6599"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - 4 \, {y} = -5 \, {x} \hspace{2em}x(0)= -3" alt="-{x'} - 4 \, {y} = -5 \, {x} \hspace{2em}x(0)= -3" title="-{x'} - 4 \, {y} = -5 \, {x} \hspace{2em}x(0)= -3" data-latex="-{x'} - 4 \, {y} = -5 \, {x} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" alt="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" title="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" data-latex="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20-%204%20%5C,%20%7By%7D%20=%20-5%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{x'} - 4 \, {y} = -5 \, {x} \hspace{2em}x(0)= -3" title="-{x'} - 4 \, {y} = -5 \, {x} \hspace{2em}x(0)= -3" data-latex="-{x'} - 4 \, {y} = -5 \, {x} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20%7Bx%7D%20=%208%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" title="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" data-latex="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0382" title="S2 | Linear ODE system with elimination | ver. 0382"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {x'} + 2 \, {x} \hspace{2em}x(0)= -2" alt="{y} = {x'} + 2 \, {x} \hspace{2em}x(0)= -2" title="{y} = {x'} + 2 \, {x} \hspace{2em}x(0)= -2" data-latex="{y} = {x'} + 2 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -5" alt="-4 \, {x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -5" title="-4 \, {x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7Bx'%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{y} = {x'} + 2 \, {x} \hspace{2em}x(0)= -2" title="{y} = {x'} + 2 \, {x} \hspace{2em}x(0)= -2" data-latex="{y} = {x'} + 2 \, {x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20-%203%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-4 \, {x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -5" title="-4 \, {x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" alt="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1382" title="S2 | Linear ODE system with elimination | ver. 1382"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -1" alt="2 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -1" title="2 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = -9 \, {y} + {y'} \hspace{2em}x(0)= 1" alt="-2 \, {x} = -9 \, {y} + {y'} \hspace{2em}x(0)= 1" title="-2 \, {x} = -9 \, {y} + {y'} \hspace{2em}x(0)= 1" data-latex="-2 \, {x} = -9 \, {y} + {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="2 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -1" title="2 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20-9%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-2 \, {x} = -9 \, {y} + {y'} \hspace{2em}x(0)= 1" title="-2 \, {x} = -9 \, {y} + {y'} \hspace{2em}x(0)= 1" data-latex="-2 \, {x} = -9 \, {y} + {y'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" alt="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7726" title="S2 | Linear ODE system with elimination | ver. 7726"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" alt="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" title="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" alt="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" title="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20=%20-%7Bx'%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" title="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20+%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" title="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-2893" title="S2 | Linear ODE system with elimination | ver. 2893"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} + {y} = {x} \hspace{2em}x(0)= 0" alt="{x'} + {y} = {x} \hspace{2em}x(0)= 0" title="{x'} + {y} = {x} \hspace{2em}x(0)= 0" data-latex="{x'} + {y} = {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -2 \, {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 5" alt="0 = -2 \, {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 5" title="0 = -2 \, {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = -2 \, {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20+%20%7By%7D%20=%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{x'} + {y} = {x} \hspace{2em}x(0)= 0" title="{x'} + {y} = {x} \hspace{2em}x(0)= 0" data-latex="{x'} + {y} = {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-2%20%5C,%20%7By%7D%20-%20%7By'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="0 = -2 \, {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 5" title="0 = -2 \, {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = -2 \, {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="x= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5694" title="S2 | Linear ODE system with elimination | ver. 5694"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -3" alt="4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -3" title="4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -3" data-latex="4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" alt="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" title="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" data-latex="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20-%205%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -3" title="4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -3" data-latex="4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20%7Bx%7D%20=%208%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" title="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2" data-latex="{y'} + {x} = 8 \, {y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7926" title="S2 | Linear ODE system with elimination | ver. 7926"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - {x} = 3 \, {y} \hspace{2em}x(0)= -2" alt="{x'} - {x} = 3 \, {y} \hspace{2em}x(0)= -2" title="{x'} - {x} = 3 \, {y} \hspace{2em}x(0)= -2" data-latex="{x'} - {x} = 3 \, {y} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {x} = -4 \, {y} - {y'} \hspace{2em}x(0)= -7" alt="-12 \, {x} = -4 \, {y} - {y'} \hspace{2em}x(0)= -7" title="-12 \, {x} = -4 \, {y} - {y'} \hspace{2em}x(0)= -7" data-latex="-12 \, {x} = -4 \, {y} - {y'} \hspace{2em}x(0)= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20-%20%7Bx%7D%20=%203%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x'} - {x} = 3 \, {y} \hspace{2em}x(0)= -2" title="{x'} - {x} = 3 \, {y} \hspace{2em}x(0)= -2" data-latex="{x'} - {x} = 3 \, {y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7Bx%7D%20=%20-4%20%5C,%20%7By%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-7" alt="-12 \, {x} = -4 \, {y} - {y'} \hspace{2em}x(0)= -7" title="-12 \, {x} = -4 \, {y} - {y'} \hspace{2em}x(0)= -7" data-latex="-12 \, {x} = -4 \, {y} - {y'} \hspace{2em}x(0)= -7"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" alt="x= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" alt="y= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="y= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="y= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-3%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="y= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="y= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5176" title="S2 | Linear ODE system with elimination | ver. 5176"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 5" alt="4 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 5" title="4 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 5" data-latex="4 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + {x} = 2 \, {y} \hspace{2em}x(0)= 0" alt="{y'} + {x} = 2 \, {y} \hspace{2em}x(0)= 0" title="{y'} + {x} = 2 \, {y} \hspace{2em}x(0)= 0" data-latex="{y'} + {x} = 2 \, {y} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20+%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="4 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 5" title="4 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 5" data-latex="4 \, {y} + {x} = -{x'} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20%7Bx%7D%20=%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{y'} + {x} = 2 \, {y} \hspace{2em}x(0)= 0" title="{y'} + {x} = 2 \, {y} \hspace{2em}x(0)= 0" data-latex="{y'} + {x} = 2 \, {y} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7301" title="S2 | Linear ODE system with elimination | ver. 7301"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -4 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" alt="{x'} = -4 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" title="{x'} = -4 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="{x'} = -4 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = -{x} + 9 \, {y} \hspace{2em}x(0)= 0" alt="-{y'} = -{x} + 9 \, {y} \hspace{2em}x(0)= 0" title="-{y'} = -{x} + 9 \, {y} \hspace{2em}x(0)= 0" data-latex="-{y'} = -{x} + 9 \, {y} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20=%20-4%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="{x'} = -4 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" title="{x'} = -4 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="{x'} = -4 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%20-%7Bx%7D%20+%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y'} = -{x} + 9 \, {y} \hspace{2em}x(0)= 0" title="-{y'} = -{x} + 9 \, {y} \hspace{2em}x(0)= 0" data-latex="-{y'} = -{x} + 9 \, {y} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" alt="x= -4 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" alt="y= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8634" title="S2 | Linear ODE system with elimination | ver. 8634"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= -2" alt="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= -2" title="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= -2" data-latex="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 3" alt="0 = {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 3" title="0 = {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 3" data-latex="0 = {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20+%202%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= -2" title="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= -2" data-latex="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7By%7D%20-%20%7By'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 3" title="0 = {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 3" data-latex="0 = {y} - {y'} - 4 \, {x} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7038" title="S2 | Linear ODE system with elimination | ver. 7038"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} = {x'} - 2 \, {x} \hspace{2em}x(0)= 0" alt="-{y} = {x'} - 2 \, {x} \hspace{2em}x(0)= 0" title="-{y} = {x'} - 2 \, {x} \hspace{2em}x(0)= 0" data-latex="-{y} = {x'} - 2 \, {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y} - {y'} - 4 \, {x} \hspace{2em}x(0)= -5" alt="0 = -{y} - {y'} - 4 \, {x} \hspace{2em}x(0)= -5" title="0 = -{y} - {y'} - 4 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = -{y} - {y'} - 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By%7D%20=%20%7Bx'%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y} = {x'} - 2 \, {x} \hspace{2em}x(0)= 0" title="-{y} = {x'} - 2 \, {x} \hspace{2em}x(0)= 0" data-latex="-{y} = {x'} - 2 \, {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By%7D%20-%20%7By'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = -{y} - {y'} - 4 \, {x} \hspace{2em}x(0)= -5" title="0 = -{y} - {y'} - 4 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = -{y} - {y'} - 4 \, {x} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6468" title="S2 | Linear ODE system with elimination | ver. 6468"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -3" alt="-{x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -3" title="-{x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -3" data-latex="-{x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 2" alt="{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 2" title="{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 2" data-latex="{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20=%20-4%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -3" title="-{x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -3" data-latex="-{x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%7Bx%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 2" title="{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 2" data-latex="{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8676" title="S2 | Linear ODE system with elimination | ver. 8676"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= 3" alt="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= 3" title="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2" alt="2 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2" title="2 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2" data-latex="2 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx%7D%20-%20%7Bx'%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= 3" title="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20-%20%7By'%7D%20=%20-%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="2 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2" title="2 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2" data-latex="2 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" alt="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7001" title="S2 | Linear ODE system with elimination | ver. 7001"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - 12 \, {y} = -2 \, {x} \hspace{2em}x(0)= -7" alt="-{x'} - 12 \, {y} = -2 \, {x} \hspace{2em}x(0)= -7" title="-{x'} - 12 \, {y} = -2 \, {x} \hspace{2em}x(0)= -7" data-latex="-{x'} - 12 \, {y} = -2 \, {x} \hspace{2em}x(0)= -7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y} = {y'} + 3 \, {x} \hspace{2em}x(0)= 2" alt="7 \, {y} = {y'} + 3 \, {x} \hspace{2em}x(0)= 2" title="7 \, {y} = {y'} + 3 \, {x} \hspace{2em}x(0)= 2" data-latex="7 \, {y} = {y'} + 3 \, {x} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20-%2012%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-7" alt="-{x'} - 12 \, {y} = -2 \, {x} \hspace{2em}x(0)= -7" title="-{x'} - 12 \, {y} = -2 \, {x} \hspace{2em}x(0)= -7" data-latex="-{x'} - 12 \, {y} = -2 \, {x} \hspace{2em}x(0)= -7"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7%20%5C,%20%7By%7D%20=%20%7By'%7D%20+%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="7 \, {y} = {y'} + 3 \, {x} \hspace{2em}x(0)= 2" title="7 \, {y} = {y'} + 3 \, {x} \hspace{2em}x(0)= 2" data-latex="7 \, {y} = {y'} + 3 \, {x} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" alt="x= -4 \, e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="x= -4 \, e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="x= -4 \, e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="x= -4 \, e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="x= -4 \, e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0353" title="S2 | Linear ODE system with elimination | ver. 0353"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2" alt="{x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2" title="{x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2" data-latex="{x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5" alt="{y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5" title="{y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5" data-latex="{y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20-%20%7By%7D%20=%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2" title="{x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2" data-latex="{x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%203%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%205" alt="{y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5" title="{y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5" data-latex="{y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(-2 \, t\right)} - e^{t}" alt="x= -e^{\left(-2 \, t\right)} - e^{t}" title="x= -e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= -e^{\left(-2 \, t\right)} - e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(-2 \, t\right)} + e^{t}" alt="y= 4 \, e^{\left(-2 \, t\right)} + e^{t}" title="y= 4 \, e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= 4 \, e^{\left(-2 \, t\right)} + e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="x= -e^{\left(-2 \, t\right)} - e^{t}" title="x= -e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= -e^{\left(-2 \, t\right)} - e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= 4 \, e^{\left(-2 \, t\right)} + e^{t}" title="y= 4 \, e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= 4 \, e^{\left(-2 \, t\right)} + e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5375" title="S2 | Linear ODE system with elimination | ver. 5375"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" alt="4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = 2 \, {x} + 9 \, {y} \hspace{2em}x(0)= 1" alt="-{y'} = 2 \, {x} + 9 \, {y} \hspace{2em}x(0)= 1" title="-{y'} = 2 \, {x} + 9 \, {y} \hspace{2em}x(0)= 1" data-latex="-{y'} = 2 \, {x} + 9 \, {y} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%202%20%5C,%20%7Bx%7D%20+%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-{y'} = 2 \, {x} + 9 \, {y} \hspace{2em}x(0)= 1" title="-{y'} = 2 \, {x} + 9 \, {y} \hspace{2em}x(0)= 1" data-latex="-{y'} = 2 \, {x} + 9 \, {y} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" alt="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-2290" title="S2 | Linear ODE system with elimination | ver. 2290"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -3" alt="-{x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -3" title="-{x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -3" data-latex="-{x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = {x} + {y'} \hspace{2em}x(0)= -2" alt="2 \, {y} = {x} + {y'} \hspace{2em}x(0)= -2" title="2 \, {y} = {x} + {y'} \hspace{2em}x(0)= -2" data-latex="2 \, {y} = {x} + {y'} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20-%204%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -3" title="-{x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -3" data-latex="-{x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="2 \, {y} = {x} + {y'} \hspace{2em}x(0)= -2" title="2 \, {y} = {x} + {y'} \hspace{2em}x(0)= -2" data-latex="2 \, {y} = {x} + {y'} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7326" title="S2 | Linear ODE system with elimination | ver. 7326"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -3" alt="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -3" title="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -3" data-latex="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= 3" alt="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= 3" title="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= 3" data-latex="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -3" title="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -3" data-latex="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20=%20-9%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= 3" title="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= 3" data-latex="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0188" title="S2 | Linear ODE system with elimination | ver. 0188"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {x} = 2 \, {y} + {x'} \hspace{2em}x(0)= -3" alt="5 \, {x} = 2 \, {y} + {x'} \hspace{2em}x(0)= -3" title="5 \, {x} = 2 \, {y} + {x'} \hspace{2em}x(0)= -3" data-latex="5 \, {x} = 2 \, {y} + {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -8 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" alt="0 = -8 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" title="0 = -8 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" data-latex="0 = -8 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="5 \, {x} = 2 \, {y} + {x'} \hspace{2em}x(0)= -3" title="5 \, {x} = 2 \, {y} + {x'} \hspace{2em}x(0)= -3" data-latex="5 \, {x} = 2 \, {y} + {x'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-8%20%5C,%20%7By%7D%20+%202%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = -8 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" title="0 = -8 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" data-latex="0 = -8 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}" alt="x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}" title="x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}" data-latex="x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" alt="y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}" title="x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}" data-latex="x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7914" title="S2 | Linear ODE system with elimination | ver. 7914"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 4 \, {y} + {x} - {x'} \hspace{2em}x(0)= -5" alt="0 = 4 \, {y} + {x} - {x'} \hspace{2em}x(0)= -5" title="0 = 4 \, {y} + {x} - {x'} \hspace{2em}x(0)= -5" data-latex="0 = 4 \, {y} + {x} - {x'} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = 4 \, {y} - 9 \, {x} \hspace{2em}x(0)= 8" alt="-{y'} = 4 \, {y} - 9 \, {x} \hspace{2em}x(0)= 8" title="-{y'} = 4 \, {y} - 9 \, {x} \hspace{2em}x(0)= 8" data-latex="-{y'} = 4 \, {y} - 9 \, {x} \hspace{2em}x(0)= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%204%20%5C,%20%7By%7D%20+%20%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = 4 \, {y} + {x} - {x'} \hspace{2em}x(0)= -5" title="0 = 4 \, {y} + {x} - {x'} \hspace{2em}x(0)= -5" data-latex="0 = 4 \, {y} + {x} - {x'} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%204%20%5C,%20%7By%7D%20-%209%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%208" alt="-{y'} = 4 \, {y} - 9 \, {x} \hspace{2em}x(0)= 8" title="-{y'} = 4 \, {y} - 9 \, {x} \hspace{2em}x(0)= 8" data-latex="-{y'} = 4 \, {y} - 9 \, {x} \hspace{2em}x(0)= 8"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" alt="x= -e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="x= -e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="x= -e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" alt="y= -e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="x= -e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="x= -e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%209%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8896" title="S2 | Linear ODE system with elimination | ver. 8896"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x'} - 4 \, {y} + 5 \, {x} \hspace{2em}x(0)= -3" alt="0 = -{x'} - 4 \, {y} + 5 \, {x} \hspace{2em}x(0)= -3" title="0 = -{x'} - 4 \, {y} + 5 \, {x} \hspace{2em}x(0)= -3" data-latex="0 = -{x'} - 4 \, {y} + 5 \, {x} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8 \, {y} - {x} = {y'} \hspace{2em}x(0)= -2" alt="8 \, {y} - {x} = {y'} \hspace{2em}x(0)= -2" title="8 \, {y} - {x} = {y'} \hspace{2em}x(0)= -2" data-latex="8 \, {y} - {x} = {y'} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx'%7D%20-%204%20%5C,%20%7By%7D%20+%205%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = -{x'} - 4 \, {y} + 5 \, {x} \hspace{2em}x(0)= -3" title="0 = -{x'} - 4 \, {y} + 5 \, {x} \hspace{2em}x(0)= -3" data-latex="0 = -{x'} - 4 \, {y} + 5 \, {x} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8%20%5C,%20%7By%7D%20-%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="8 \, {y} - {x} = {y'} \hspace{2em}x(0)= -2" title="8 \, {y} - {x} = {y'} \hspace{2em}x(0)= -2" data-latex="8 \, {y} - {x} = {y'} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-2864" title="S2 | Linear ODE system with elimination | ver. 2864"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {y} + {x'} = -{x} \hspace{2em}x(0)= -7" alt="12 \, {y} + {x'} = -{x} \hspace{2em}x(0)= -7" title="12 \, {y} + {x'} = -{x} \hspace{2em}x(0)= -7" data-latex="12 \, {y} + {x'} = -{x} \hspace{2em}x(0)= -7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -4 \, {y} + 3 \, {x} + {y'} \hspace{2em}x(0)= 2" alt="0 = -4 \, {y} + 3 \, {x} + {y'} \hspace{2em}x(0)= 2" title="0 = -4 \, {y} + 3 \, {x} + {y'} \hspace{2em}x(0)= 2" data-latex="0 = -4 \, {y} + 3 \, {x} + {y'} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20=%20-%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-7" alt="12 \, {y} + {x'} = -{x} \hspace{2em}x(0)= -7" title="12 \, {y} + {x'} = -{x} \hspace{2em}x(0)= -7" data-latex="12 \, {y} + {x'} = -{x} \hspace{2em}x(0)= -7"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-4%20%5C,%20%7By%7D%20+%203%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="0 = -4 \, {y} + 3 \, {x} + {y'} \hspace{2em}x(0)= 2" title="0 = -4 \, {y} + 3 \, {x} + {y'} \hspace{2em}x(0)= 2" data-latex="0 = -4 \, {y} + 3 \, {x} + {y'} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" alt="x= -4 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="x= -4 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="x= -4 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" alt="y= 3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="y= 3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="y= 3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="x= -4 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="x= -4 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="y= 3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="y= 3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6818" title="S2 | Linear ODE system with elimination | ver. 6818"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5" alt="6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5" title="6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5" data-latex="6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= 1" alt="4 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= 1" title="4 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= 1" data-latex="4 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By%7D%20=%20-%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5" title="6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5" data-latex="6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20-%206%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="4 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= 1" title="4 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= 1" data-latex="4 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" alt="x= -2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="x= -2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="x= -2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" alt="y= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="y= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="y= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-2%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="x= -2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="x= -2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="y= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="y= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1301" title="S2 | Linear ODE system with elimination | ver. 1301"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} + {x'} = -2 \, {y} \hspace{2em}x(0)= 3" alt="-{x} + {x'} = -2 \, {y} \hspace{2em}x(0)= 3" title="-{x} + {x'} = -2 \, {y} \hspace{2em}x(0)= 3" data-latex="-{x} + {x'} = -2 \, {y} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} - {y'} = 2 \, {x} \hspace{2em}x(0)= 1" alt="-2 \, {y} - {y'} = 2 \, {x} \hspace{2em}x(0)= 1" title="-2 \, {y} - {y'} = 2 \, {x} \hspace{2em}x(0)= 1" data-latex="-2 \, {y} - {y'} = 2 \, {x} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20+%20%7Bx'%7D%20=%20-2%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-{x} + {x'} = -2 \, {y} \hspace{2em}x(0)= 3" title="-{x} + {x'} = -2 \, {y} \hspace{2em}x(0)= 3" data-latex="-{x} + {x'} = -2 \, {y} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20-%20%7By'%7D%20=%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-2 \, {y} - {y'} = 2 \, {x} \hspace{2em}x(0)= 1" title="-2 \, {y} - {y'} = 2 \, {x} \hspace{2em}x(0)= 1" data-latex="-2 \, {y} - {y'} = 2 \, {x} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" alt="y= -e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-9018" title="S2 | Linear ODE system with elimination | ver. 9018"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = -{x'} - 9 \, {y} \hspace{2em}x(0)= 10" alt="-4 \, {x} = -{x'} - 9 \, {y} \hspace{2em}x(0)= 10" title="-4 \, {x} = -{x'} - 9 \, {y} \hspace{2em}x(0)= 10" data-latex="-4 \, {x} = -{x'} - 9 \, {y} \hspace{2em}x(0)= 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - {y} = 4 \, {x} \hspace{2em}x(0)= -3" alt="-{y'} - {y} = 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} - {y} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} - {y} = 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20-%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%2010" alt="-4 \, {x} = -{x'} - 9 \, {y} \hspace{2em}x(0)= 10" title="-4 \, {x} = -{x'} - 9 \, {y} \hspace{2em}x(0)= 10" data-latex="-4 \, {x} = -{x'} - 9 \, {y} \hspace{2em}x(0)= 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20-%20%7By%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} - {y} = 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} - {y} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} - {y} = 4 \, {x} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" alt="x= 9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" alt="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%209%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4506" title="S2 | Linear ODE system with elimination | ver. 4506"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} = -{x} + {x'} \hspace{2em}x(0)= 5" alt="-6 \, {y} = -{x} + {x'} \hspace{2em}x(0)= 5" title="-6 \, {y} = -{x} + {x'} \hspace{2em}x(0)= 5" data-latex="-6 \, {y} = -{x} + {x'} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {x} - 4 \, {y} = {y'} \hspace{2em}x(0)= 1" alt="-6 \, {x} - 4 \, {y} = {y'} \hspace{2em}x(0)= 1" title="-6 \, {x} - 4 \, {y} = {y'} \hspace{2em}x(0)= 1" data-latex="-6 \, {x} - 4 \, {y} = {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20=%20-%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-6 \, {y} = -{x} + {x'} \hspace{2em}x(0)= 5" title="-6 \, {y} = -{x} + {x'} \hspace{2em}x(0)= 5" data-latex="-6 \, {y} = -{x} + {x'} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-6 \, {x} - 4 \, {y} = {y'} \hspace{2em}x(0)= 1" title="-6 \, {x} - 4 \, {y} = {y'} \hspace{2em}x(0)= 1" data-latex="-6 \, {x} - 4 \, {y} = {y'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" alt="x= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="x= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="x= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}" alt="y= -2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="x= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="x= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1389" title="S2 | Linear ODE system with elimination | ver. 1389"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} - {x} = -{x'} \hspace{2em}x(0)= 0" alt="-{y} - {x} = -{x'} \hspace{2em}x(0)= 0" title="-{y} - {x} = -{x'} \hspace{2em}x(0)= 0" data-latex="-{y} - {x} = -{x'} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" alt="-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" title="-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" data-latex="-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By%7D%20-%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y} - {x} = -{x'} \hspace{2em}x(0)= 0" title="-{y} - {x} = -{x'} \hspace{2em}x(0)= 0" data-latex="-{y} - {x} = -{x'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" title="-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" data-latex="-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" alt="x= e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" alt="y= 4 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" title="y= 4 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" data-latex="y= 4 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" title="y= 4 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" data-latex="y= 4 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6941" title="S2 | Linear ODE system with elimination | ver. 6941"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -{x'} + 3 \, {x} \hspace{2em}x(0)= 3" alt="2 \, {y} = -{x'} + 3 \, {x} \hspace{2em}x(0)= 3" title="2 \, {y} = -{x'} + 3 \, {x} \hspace{2em}x(0)= 3" data-latex="2 \, {y} = -{x'} + 3 \, {x} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + 2 \, {x} = -8 \, {y} \hspace{2em}x(0)= -3" alt="-{y'} + 2 \, {x} = -8 \, {y} \hspace{2em}x(0)= -3" title="-{y'} + 2 \, {x} = -8 \, {y} \hspace{2em}x(0)= -3" data-latex="-{y'} + 2 \, {x} = -8 \, {y} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20+%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="2 \, {y} = -{x'} + 3 \, {x} \hspace{2em}x(0)= 3" title="2 \, {y} = -{x'} + 3 \, {x} \hspace{2em}x(0)= 3" data-latex="2 \, {y} = -{x'} + 3 \, {x} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20=%20-8%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} + 2 \, {x} = -8 \, {y} \hspace{2em}x(0)= -3" title="-{y'} + 2 \, {x} = -8 \, {y} \hspace{2em}x(0)= -3" data-latex="-{y'} + 2 \, {x} = -8 \, {y} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" alt="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" alt="y= -2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4225" title="S2 | Linear ODE system with elimination | ver. 4225"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 1" alt="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 1" title="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 1" data-latex="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= 3" alt="7 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= 3" title="7 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= 3" data-latex="7 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 1" title="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 1" data-latex="-2 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="7 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= 3" title="7 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= 3" data-latex="7 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4298" title="S2 | Linear ODE system with elimination | ver. 4298"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" alt="{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" title="{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" data-latex="{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {x} + {y} = {y'} \hspace{2em}x(0)= -1" alt="6 \, {x} + {y} = {y'} \hspace{2em}x(0)= -1" title="6 \, {x} + {y} = {y'} \hspace{2em}x(0)= -1" data-latex="6 \, {x} + {y} = {y'} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20-%206%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" title="{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" data-latex="{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7Bx%7D%20+%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="6 \, {x} + {y} = {y'} \hspace{2em}x(0)= -1" title="6 \, {x} + {y} = {y'} \hspace{2em}x(0)= -1" data-latex="6 \, {x} + {y} = {y'} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" alt="x= -2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= -2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= -2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" alt="y= -3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-2%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= -2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= -2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-3%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6874" title="S2 | Linear ODE system with elimination | ver. 6874"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {y} = -{x'} + 2 \, {x} \hspace{2em}x(0)= 7" alt="-12 \, {y} = -{x'} + 2 \, {x} \hspace{2em}x(0)= 7" title="-12 \, {y} = -{x'} + 2 \, {x} \hspace{2em}x(0)= 7" data-latex="-12 \, {y} = -{x'} + 2 \, {x} \hspace{2em}x(0)= 7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -3 \, {x} - 7 \, {y} + {y'} \hspace{2em}x(0)= 2" alt="0 = -3 \, {x} - 7 \, {y} + {y'} \hspace{2em}x(0)= 2" title="0 = -3 \, {x} - 7 \, {y} + {y'} \hspace{2em}x(0)= 2" data-latex="0 = -3 \, {x} - 7 \, {y} + {y'} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="-12 \, {y} = -{x'} + 2 \, {x} \hspace{2em}x(0)= 7" title="-12 \, {y} = -{x'} + 2 \, {x} \hspace{2em}x(0)= 7" data-latex="-12 \, {y} = -{x'} + 2 \, {x} \hspace{2em}x(0)= 7"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7Bx%7D%20-%207%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="0 = -3 \, {x} - 7 \, {y} + {y'} \hspace{2em}x(0)= 2" title="0 = -3 \, {x} - 7 \, {y} + {y'} \hspace{2em}x(0)= 2" data-latex="0 = -3 \, {x} - 7 \, {y} + {y'} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 4 \, e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}" alt="x= 4 \, e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}" title="x= 4 \, e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}" data-latex="x= 4 \, e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%204%20%5C,%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= 4 \, e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}" title="x= 4 \, e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}" data-latex="x= 4 \, e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= 3 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8870" title="S2 | Linear ODE system with elimination | ver. 8870"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y} + {x'} - 3 \, {x} \hspace{2em}x(0)= 0" alt="0 = -{y} + {x'} - 3 \, {x} \hspace{2em}x(0)= 0" title="0 = -{y} + {x'} - 3 \, {x} \hspace{2em}x(0)= 0" data-latex="0 = -{y} + {x'} - 3 \, {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = 2 \, {y} + {y'} \hspace{2em}x(0)= -3" alt="-4 \, {x} = 2 \, {y} + {y'} \hspace{2em}x(0)= -3" title="-4 \, {x} = 2 \, {y} + {y'} \hspace{2em}x(0)= -3" data-latex="-4 \, {x} = 2 \, {y} + {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By%7D%20+%20%7Bx'%7D%20-%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="0 = -{y} + {x'} - 3 \, {x} \hspace{2em}x(0)= 0" title="0 = -{y} + {x'} - 3 \, {x} \hspace{2em}x(0)= 0" data-latex="0 = -{y} + {x'} - 3 \, {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-4 \, {x} = 2 \, {y} + {y'} \hspace{2em}x(0)= -3" title="-4 \, {x} = 2 \, {y} + {y'} \hspace{2em}x(0)= -3" data-latex="-4 \, {x} = 2 \, {y} + {y'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" alt="x= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}" alt="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}" title="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}" title="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1046" title="S2 | Linear ODE system with elimination | ver. 1046"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 5 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -3" alt="0 = 5 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -3" title="0 = 5 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -3" data-latex="0 = 5 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} - 8 \, {y} + {y'} = 0 \hspace{2em}x(0)= -2" alt="{x} - 8 \, {y} + {y'} = 0 \hspace{2em}x(0)= -2" title="{x} - 8 \, {y} + {y'} = 0 \hspace{2em}x(0)= -2" data-latex="{x} - 8 \, {y} + {y'} = 0 \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%205%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = 5 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -3" title="0 = 5 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -3" data-latex="0 = 5 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20-%208%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x} - 8 \, {y} + {y'} = 0 \hspace{2em}x(0)= -2" title="{x} - 8 \, {y} + {y'} = 0 \hspace{2em}x(0)= -2" data-latex="{x} - 8 \, {y} + {y'} = 0 \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" title="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0364" title="S2 | Linear ODE system with elimination | ver. 0364"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + {y} = {x} \hspace{2em}x(0)= 0" alt="-{x'} + {y} = {x} \hspace{2em}x(0)= 0" title="-{x'} + {y} = {x} \hspace{2em}x(0)= 0" data-latex="-{x'} + {y} = {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - 6 \, {y} - 4 \, {x} = 0 \hspace{2em}x(0)= -3" alt="-{y'} - 6 \, {y} - 4 \, {x} = 0 \hspace{2em}x(0)= -3" title="-{y'} - 6 \, {y} - 4 \, {x} = 0 \hspace{2em}x(0)= -3" data-latex="-{y'} - 6 \, {y} - 4 \, {x} = 0 \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20+%20%7By%7D%20=%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x'} + {y} = {x} \hspace{2em}x(0)= 0" title="-{x'} + {y} = {x} \hspace{2em}x(0)= 0" data-latex="-{x'} + {y} = {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20-%206%20%5C,%20%7By%7D%20-%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} - 6 \, {y} - 4 \, {x} = 0 \hspace{2em}x(0)= -3" title="-{y'} - 6 \, {y} - 4 \, {x} = 0 \hspace{2em}x(0)= -3" data-latex="-{y'} - 6 \, {y} - 4 \, {x} = 0 \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" alt="x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6445" title="S2 | Linear ODE system with elimination | ver. 6445"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -3 \, {x} - 6 \, {y} \hspace{2em}x(0)= -5" alt="-{x'} = -3 \, {x} - 6 \, {y} \hspace{2em}x(0)= -5" title="-{x'} = -3 \, {x} - 6 \, {y} \hspace{2em}x(0)= -5" data-latex="-{x'} = -3 \, {x} - 6 \, {y} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {x} - 8 \, {y} = -{y'} \hspace{2em}x(0)= -1" alt="-6 \, {x} - 8 \, {y} = -{y'} \hspace{2em}x(0)= -1" title="-6 \, {x} - 8 \, {y} = -{y'} \hspace{2em}x(0)= -1" data-latex="-6 \, {x} - 8 \, {y} = -{y'} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-3%20%5C,%20%7Bx%7D%20-%206%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-{x'} = -3 \, {x} - 6 \, {y} \hspace{2em}x(0)= -5" title="-{x'} = -3 \, {x} - 6 \, {y} \hspace{2em}x(0)= -5" data-latex="-{x'} = -3 \, {x} - 6 \, {y} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7Bx%7D%20-%208%20%5C,%20%7By%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-6 \, {x} - 8 \, {y} = -{y'} \hspace{2em}x(0)= -1" title="-6 \, {x} - 8 \, {y} = -{y'} \hspace{2em}x(0)= -1" data-latex="-6 \, {x} - 8 \, {y} = -{y'} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -2 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}" alt="x= -2 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}" title="x= -2 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}" data-latex="x= -2 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -3 \, e^{\left(12 \, t\right)} + 2 \, e^{\left(-t\right)}" alt="y= -3 \, e^{\left(12 \, t\right)} + 2 \, e^{\left(-t\right)}" title="y= -3 \, e^{\left(12 \, t\right)} + 2 \, e^{\left(-t\right)}" data-latex="y= -3 \, e^{\left(12 \, t\right)} + 2 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-2%20%5C,%20e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= -2 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}" title="x= -2 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}" data-latex="x= -2 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-3%20%5C,%20e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -3 \, e^{\left(12 \, t\right)} + 2 \, e^{\left(-t\right)}" title="y= -3 \, e^{\left(12 \, t\right)} + 2 \, e^{\left(-t\right)}" data-latex="y= -3 \, e^{\left(12 \, t\right)} + 2 \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8419" title="S2 | Linear ODE system with elimination | ver. 8419"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= 0" alt="-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= 0" title="-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= 0" data-latex="-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= -5" alt="2 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= -5" title="2 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= -5" data-latex="2 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-%7By%7D%20-%205%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= 0" title="-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= 0" data-latex="-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="2 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= -5" title="2 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= -5" data-latex="2 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(6 \, t\right)} + e^{t}" alt="x= -e^{\left(6 \, t\right)} + e^{t}" title="x= -e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -e^{\left(6 \, t\right)} + e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(6 \, t\right)} - 4 \, e^{t}" alt="y= -e^{\left(6 \, t\right)} - 4 \, e^{t}" title="y= -e^{\left(6 \, t\right)} - 4 \, e^{t}" data-latex="y= -e^{\left(6 \, t\right)} - 4 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="x= -e^{\left(6 \, t\right)} + e^{t}" title="x= -e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -e^{\left(6 \, t\right)} + e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7Bt%7D" alt="y= -e^{\left(6 \, t\right)} - 4 \, e^{t}" title="y= -e^{\left(6 \, t\right)} - 4 \, e^{t}" data-latex="y= -e^{\left(6 \, t\right)} - 4 \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4568" title="S2 | Linear ODE system with elimination | ver. 4568"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= 10" alt="-9 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= 10" title="-9 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= 10" data-latex="-9 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" alt="-{y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" title="-{y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%2010" alt="-9 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= 10" title="-9 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= 10" data-latex="-9 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By%7D%20+%20%7By'%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" title="-{y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" alt="x= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" alt="y= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="y= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="y= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%209%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="y= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="y= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3128" title="S2 | Linear ODE system with elimination | ver. 3128"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} + {x} - {x'} = 0 \hspace{2em}x(0)= 3" alt="2 \, {y} + {x} - {x'} = 0 \hspace{2em}x(0)= 3" title="2 \, {y} + {x} - {x'} = 0 \hspace{2em}x(0)= 3" data-latex="2 \, {y} + {x} - {x'} = 0 \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} + {y'} = -2 \, {x} \hspace{2em}x(0)= 3" alt="-6 \, {y} + {y'} = -2 \, {x} \hspace{2em}x(0)= 3" title="-6 \, {y} + {y'} = -2 \, {x} \hspace{2em}x(0)= 3" data-latex="-6 \, {y} + {y'} = -2 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20+%20%7Bx%7D%20-%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%203" alt="2 \, {y} + {x} - {x'} = 0 \hspace{2em}x(0)= 3" title="2 \, {y} + {x} - {x'} = 0 \hspace{2em}x(0)= 3" data-latex="2 \, {y} + {x} - {x'} = 0 \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%20-2%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-6 \, {y} + {y'} = -2 \, {x} \hspace{2em}x(0)= 3" title="-6 \, {y} + {y'} = -2 \, {x} \hspace{2em}x(0)= 3" data-latex="-6 \, {y} + {y'} = -2 \, {x} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" alt="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" alt="y= 2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= 2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= 2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= 2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= 2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1883" title="S2 | Linear ODE system with elimination | ver. 1883"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {x} - {x'} = 2 \, {y} \hspace{2em}x(0)= 1" alt="5 \, {x} - {x'} = 2 \, {y} \hspace{2em}x(0)= 1" title="5 \, {x} - {x'} = 2 \, {y} \hspace{2em}x(0)= 1" data-latex="5 \, {x} - {x'} = 2 \, {y} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = -8 \, {y} + {y'} \hspace{2em}x(0)= 3" alt="-2 \, {x} = -8 \, {y} + {y'} \hspace{2em}x(0)= 3" title="-2 \, {x} = -8 \, {y} + {y'} \hspace{2em}x(0)= 3" data-latex="-2 \, {x} = -8 \, {y} + {y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20=%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="5 \, {x} - {x'} = 2 \, {y} \hspace{2em}x(0)= 1" title="5 \, {x} - {x'} = 2 \, {y} \hspace{2em}x(0)= 1" data-latex="5 \, {x} - {x'} = 2 \, {y} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20-8%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-2 \, {x} = -8 \, {y} + {y'} \hspace{2em}x(0)= 3" title="-2 \, {x} = -8 \, {y} + {y'} \hspace{2em}x(0)= 3" data-latex="-2 \, {x} = -8 \, {y} + {y'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(9 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" alt="x= -e^{\left(9 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= -e^{\left(9 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= -e^{\left(9 \, t\right)} + 2 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}" alt="y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}" title="y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(9 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= -e^{\left(9 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= -e^{\left(9 \, t\right)} + 2 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}" title="y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0387" title="S2 | Linear ODE system with elimination | ver. 0387"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -{x'} + 4 \, {x} \hspace{2em}x(0)= -1" alt="2 \, {y} = -{x'} + 4 \, {x} \hspace{2em}x(0)= -1" title="2 \, {y} = -{x'} + 4 \, {x} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = -{x'} + 4 \, {x} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= -1" alt="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= -1" title="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= -1" data-latex="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="2 \, {y} = -{x'} + 4 \, {x} \hspace{2em}x(0)= -1" title="2 \, {y} = -{x'} + 4 \, {x} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = -{x'} + 4 \, {x} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20=%20-9%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= -1" title="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= -1" data-latex="-{y'} + 2 \, {x} = -9 \, {y} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" alt="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" alt="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1824" title="S2 | Linear ODE system with elimination | ver. 1824"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y} + {x'} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" alt="6 \, {y} + {x'} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" title="6 \, {y} + {x'} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" data-latex="6 \, {y} + {x'} - 3 \, {x} = 0 \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {x} = -{y'} - 2 \, {y} \hspace{2em}x(0)= -1" alt="6 \, {x} = -{y'} - 2 \, {y} \hspace{2em}x(0)= -1" title="6 \, {x} = -{y'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="6 \, {x} = -{y'} - 2 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20-%203%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="6 \, {y} + {x'} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" title="6 \, {y} + {x'} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" data-latex="6 \, {y} + {x'} - 3 \, {x} = 0 \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="6 \, {x} = -{y'} - 2 \, {y} \hspace{2em}x(0)= -1" title="6 \, {x} = -{y'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="6 \, {x} = -{y'} - 2 \, {y} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" alt="x= -3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="x= -3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="x= -3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" alt="y= 2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="y= 2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="y= 2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-3%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="x= -3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="x= -3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="y= 2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="y= 2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6886" title="S2 | Linear ODE system with elimination | ver. 6886"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} - 9 \, {y} = -{x'} \hspace{2em}x(0)= -8" alt="{x} - 9 \, {y} = -{x'} \hspace{2em}x(0)= -8" title="{x} - 9 \, {y} = -{x'} \hspace{2em}x(0)= -8" data-latex="{x} - 9 \, {y} = -{x'} \hspace{2em}x(0)= -8"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} - 4 \, {y} + {y'} = 0 \hspace{2em}x(0)= 5" alt="-4 \, {x} - 4 \, {y} + {y'} = 0 \hspace{2em}x(0)= 5" title="-4 \, {x} - 4 \, {y} + {y'} = 0 \hspace{2em}x(0)= 5" data-latex="-4 \, {x} - 4 \, {y} + {y'} = 0 \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20-%209%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-8" alt="{x} - 9 \, {y} = -{x'} \hspace{2em}x(0)= -8" title="{x} - 9 \, {y} = -{x'} \hspace{2em}x(0)= -8" data-latex="{x} - 9 \, {y} = -{x'} \hspace{2em}x(0)= -8"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%205" alt="-4 \, {x} - 4 \, {y} + {y'} = 0 \hspace{2em}x(0)= 5" title="-4 \, {x} - 4 \, {y} + {y'} = 0 \hspace{2em}x(0)= 5" data-latex="-4 \, {x} - 4 \, {y} + {y'} = 0 \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" alt="x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5655" title="S2 | Linear ODE system with elimination | ver. 5655"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} - 2 \, {x} - {x'} = 0 \hspace{2em}x(0)= 2" alt="{y} - 2 \, {x} - {x'} = 0 \hspace{2em}x(0)= 2" title="{y} - 2 \, {x} - {x'} = 0 \hspace{2em}x(0)= 2" data-latex="{y} - 2 \, {x} - {x'} = 0 \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} - {y'} = -4 \, {x} \hspace{2em}x(0)= 3" alt="{y} - {y'} = -4 \, {x} \hspace{2em}x(0)= 3" title="{y} - {y'} = -4 \, {x} \hspace{2em}x(0)= 3" data-latex="{y} - {y'} = -4 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20-%202%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y} - 2 \, {x} - {x'} = 0 \hspace{2em}x(0)= 2" title="{y} - 2 \, {x} - {x'} = 0 \hspace{2em}x(0)= 2" data-latex="{y} - 2 \, {x} - {x'} = 0 \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20-%20%7By'%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{y} - {y'} = -4 \, {x} \hspace{2em}x(0)= 3" title="{y} - {y'} = -4 \, {x} \hspace{2em}x(0)= 3" data-latex="{y} - {y'} = -4 \, {x} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0379" title="S2 | Linear ODE system with elimination | ver. 0379"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -1" alt="2 \, {y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -1" title="2 \, {y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = 5 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" alt="-{y'} = 5 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" title="-{y'} = 5 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" data-latex="-{y'} = 5 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="2 \, {y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -1" title="2 \, {y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%205%20%5C,%20%7By%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-{y'} = 5 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" title="-{y'} = 5 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" data-latex="-{y'} = 5 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" alt="x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}" alt="y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-2%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3495" title="S2 | Linear ODE system with elimination | ver. 3495"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = 2 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" alt="-{x'} = 2 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" title="-{x'} = 2 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-{x'} = 2 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} - {x} - {y} = 0 \hspace{2em}x(0)= 2" alt="{y'} - {x} - {y} = 0 \hspace{2em}x(0)= 2" title="{y'} - {x} - {y} = 0 \hspace{2em}x(0)= 2" data-latex="{y'} - {x} - {y} = 0 \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%202%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{x'} = 2 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" title="-{x'} = 2 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-{x'} = 2 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20-%20%7Bx%7D%20-%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y'} - {x} - {y} = 0 \hspace{2em}x(0)= 2" title="{y'} - {x} - {y} = 0 \hspace{2em}x(0)= 2" data-latex="{y'} - {x} - {y} = 0 \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1034" title="S2 | Linear ODE system with elimination | ver. 1034"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -4 \, {y} - {x} + {x'} \hspace{2em}x(0)= -3" alt="0 = -4 \, {y} - {x} + {x'} \hspace{2em}x(0)= -3" title="0 = -4 \, {y} - {x} + {x'} \hspace{2em}x(0)= -3" data-latex="0 = -4 \, {y} - {x} + {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 4 \, {y} = 9 \, {x} \hspace{2em}x(0)= 10" alt="{y'} + 4 \, {y} = 9 \, {x} \hspace{2em}x(0)= 10" title="{y'} + 4 \, {y} = 9 \, {x} \hspace{2em}x(0)= 10" data-latex="{y'} + 4 \, {y} = 9 \, {x} \hspace{2em}x(0)= 10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-4%20%5C,%20%7By%7D%20-%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = -4 \, {y} - {x} + {x'} \hspace{2em}x(0)= -3" title="0 = -4 \, {y} - {x} + {x'} \hspace{2em}x(0)= -3" data-latex="0 = -4 \, {y} - {x} + {x'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%204%20%5C,%20%7By%7D%20=%209%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%2010" alt="{y'} + 4 \, {y} = 9 \, {x} \hspace{2em}x(0)= 10" title="{y'} + 4 \, {y} = 9 \, {x} \hspace{2em}x(0)= 10" data-latex="{y'} + 4 \, {y} = 9 \, {x} \hspace{2em}x(0)= 10"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" alt="x= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" alt="y= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" title="y= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" data-latex="y= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%209%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" title="y= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}" data-latex="y= e^{\left(5 \, t\right)} + 9 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8885" title="S2 | Linear ODE system with elimination | ver. 8885"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0" alt="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0" title="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0" data-latex="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} - {y'} = 5 \, {y} \hspace{2em}x(0)= -5" alt="4 \, {x} - {y'} = 5 \, {y} \hspace{2em}x(0)= -5" title="4 \, {x} - {y'} = 5 \, {y} \hspace{2em}x(0)= -5" data-latex="4 \, {x} - {y'} = 5 \, {y} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20=%20-2%20%5C,%20%7Bx%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0" title="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0" data-latex="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20=%205%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="4 \, {x} - {y'} = 5 \, {y} \hspace{2em}x(0)= -5" title="4 \, {x} - {y'} = 5 \, {y} \hspace{2em}x(0)= -5" data-latex="4 \, {x} - {y'} = 5 \, {y} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" alt="x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" alt="y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7945" title="S2 | Linear ODE system with elimination | ver. 7945"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - 2 \, {y} = {x} \hspace{2em}x(0)= 1" alt="{x'} - 2 \, {y} = {x} \hspace{2em}x(0)= 1" title="{x'} - 2 \, {y} = {x} \hspace{2em}x(0)= 1" data-latex="{x'} - 2 \, {y} = {x} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" alt="2 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" title="2 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" data-latex="2 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20-%202%20%5C,%20%7By%7D%20=%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="{x'} - 2 \, {y} = {x} \hspace{2em}x(0)= 1" title="{x'} - 2 \, {y} = {x} \hspace{2em}x(0)= 1" data-latex="{x'} - 2 \, {y} = {x} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="2 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" title="2 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" data-latex="2 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7391" title="S2 | Linear ODE system with elimination | ver. 7391"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -{y} - 2 \, {x} \hspace{2em}x(0)= 0" alt="-{x'} = -{y} - 2 \, {x} \hspace{2em}x(0)= 0" title="-{x'} = -{y} - 2 \, {x} \hspace{2em}x(0)= 0" data-latex="-{x'} = -{y} - 2 \, {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y} - 4 \, {x} + {y'} \hspace{2em}x(0)= 5" alt="0 = {y} - 4 \, {x} + {y'} \hspace{2em}x(0)= 5" title="0 = {y} - 4 \, {x} + {y'} \hspace{2em}x(0)= 5" data-latex="0 = {y} - 4 \, {x} + {y'} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-%7By%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x'} = -{y} - 2 \, {x} \hspace{2em}x(0)= 0" title="-{x'} = -{y} - 2 \, {x} \hspace{2em}x(0)= 0" data-latex="-{x'} = -{y} - 2 \, {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7By%7D%20-%204%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="0 = {y} - 4 \, {x} + {y'} \hspace{2em}x(0)= 5" title="0 = {y} - 4 \, {x} + {y'} \hspace{2em}x(0)= 5" data-latex="0 = {y} - 4 \, {x} + {y'} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" alt="y= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="y= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="y= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="y= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="y= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5727" title="S2 | Linear ODE system with elimination | ver. 5727"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} + {x} = {x'} \hspace{2em}x(0)= 0" alt="{y} + {x} = {x'} \hspace{2em}x(0)= 0" title="{y} + {x} = {x'} \hspace{2em}x(0)= 0" data-latex="{y} + {x} = {x'} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -2 \, {y} + 4 \, {x} \hspace{2em}x(0)= 5" alt="{y'} = -2 \, {y} + 4 \, {x} \hspace{2em}x(0)= 5" title="{y'} = -2 \, {y} + 4 \, {x} \hspace{2em}x(0)= 5" data-latex="{y'} = -2 \, {y} + 4 \, {x} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20+%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{y} + {x} = {x'} \hspace{2em}x(0)= 0" title="{y} + {x} = {x'} \hspace{2em}x(0)= 0" data-latex="{y} + {x} = {x'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-2%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="{y'} = -2 \, {y} + 4 \, {x} \hspace{2em}x(0)= 5" title="{y'} = -2 \, {y} + 4 \, {x} \hspace{2em}x(0)= 5" data-latex="{y'} = -2 \, {y} + 4 \, {x} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7770" title="S2 | Linear ODE system with elimination | ver. 7770"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} + {x'} + 2 \, {y} = 0 \hspace{2em}x(0)= 3" alt="{x} + {x'} + 2 \, {y} = 0 \hspace{2em}x(0)= 3" title="{x} + {x'} + 2 \, {y} = 0 \hspace{2em}x(0)= 3" data-latex="{x} + {x'} + 2 \, {y} = 0 \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -1" alt="-2 \, {x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -1" title="-2 \, {x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20+%20%7Bx'%7D%20+%202%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%203" alt="{x} + {x'} + 2 \, {y} = 0 \hspace{2em}x(0)= 3" title="{x} + {x'} + 2 \, {y} = 0 \hspace{2em}x(0)= 3" data-latex="{x} + {x'} + 2 \, {y} = 0 \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-2 \, {x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -1" title="-2 \, {x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7477" title="S2 | Linear ODE system with elimination | ver. 7477"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -2" alt="{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -2" title="{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -2" data-latex="{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} - {y} = -{y'} \hspace{2em}x(0)= 3" alt="4 \, {x} - {y} = -{y'} \hspace{2em}x(0)= 3" title="4 \, {x} - {y} = -{y'} \hspace{2em}x(0)= 3" data-latex="4 \, {x} - {y} = -{y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-%7Bx'%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -2" title="{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -2" data-latex="{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20-%20%7By%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="4 \, {x} - {y} = -{y'} \hspace{2em}x(0)= 3" title="4 \, {x} - {y} = -{y'} \hspace{2em}x(0)= 3" data-latex="4 \, {x} - {y} = -{y'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6262" title="S2 | Linear ODE system with elimination | ver. 6262"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= 2" alt="-3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= 2" title="-3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= 2" data-latex="-3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -2 \, {y} + 12 \, {x} + {y'} \hspace{2em}x(0)= 7" alt="0 = -2 \, {y} + 12 \, {x} + {y'} \hspace{2em}x(0)= 7" title="0 = -2 \, {y} + 12 \, {x} + {y'} \hspace{2em}x(0)= 7" data-latex="0 = -2 \, {y} + 12 \, {x} + {y'} \hspace{2em}x(0)= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By%7D%20-%203%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="-3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= 2" title="-3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= 2" data-latex="-3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-2%20%5C,%20%7By%7D%20+%2012%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="0 = -2 \, {y} + 12 \, {x} + {y'} \hspace{2em}x(0)= 7" title="0 = -2 \, {y} + 12 \, {x} + {y'} \hspace{2em}x(0)= 7" data-latex="0 = -2 \, {y} + 12 \, {x} + {y'} \hspace{2em}x(0)= 7"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}" alt="x= -e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}" title="x= -e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" alt="y= 3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" title="y= 3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" data-latex="y= 3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}" title="x= -e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" title="y= 3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" data-latex="y= 3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3380" title="S2 | Linear ODE system with elimination | ver. 3380"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} = 2 \, {y} - {x'} \hspace{2em}x(0)= 1" alt="-{x} = 2 \, {y} - {x'} \hspace{2em}x(0)= 1" title="-{x} = 2 \, {y} - {x'} \hspace{2em}x(0)= 1" data-latex="-{x} = 2 \, {y} - {x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= -1" alt="6 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= -1" title="6 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= -1" data-latex="6 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20=%202%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-{x} = 2 \, {y} - {x'} \hspace{2em}x(0)= 1" title="-{x} = 2 \, {y} - {x'} \hspace{2em}x(0)= 1" data-latex="-{x} = 2 \, {y} - {x'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By%7D%20=%202%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="6 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= -1" title="6 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= -1" data-latex="6 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" alt="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" alt="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6068" title="S2 | Linear ODE system with elimination | ver. 6068"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= -1" alt="-6 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= -1" title="-6 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= -1" data-latex="-6 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} - 6 \, {x} = {y} \hspace{2em}x(0)= 5" alt="{y'} - 6 \, {x} = {y} \hspace{2em}x(0)= 5" title="{y'} - 6 \, {x} = {y} \hspace{2em}x(0)= 5" data-latex="{y'} - 6 \, {x} = {y} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-6 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= -1" title="-6 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= -1" data-latex="-6 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20-%206%20%5C,%20%7Bx%7D%20=%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="{y'} - 6 \, {x} = {y} \hspace{2em}x(0)= 5" title="{y'} - 6 \, {x} = {y} \hspace{2em}x(0)= 5" data-latex="{y'} - 6 \, {x} = {y} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" alt="x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" alt="y= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3218" title="S2 | Linear ODE system with elimination | ver. 3218"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} = {x} + {x'} \hspace{2em}x(0)= -2" alt="-{y} = {x} + {x'} \hspace{2em}x(0)= -2" title="-{y} = {x} + {x'} \hspace{2em}x(0)= -2" data-latex="-{y} = {x} + {x'} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = -{y'} - 6 \, {y} \hspace{2em}x(0)= -5" alt="-4 \, {x} = -{y'} - 6 \, {y} \hspace{2em}x(0)= -5" title="-4 \, {x} = -{y'} - 6 \, {y} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} = -{y'} - 6 \, {y} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By%7D%20=%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y} = {x} + {x'} \hspace{2em}x(0)= -2" title="-{y} = {x} + {x'} \hspace{2em}x(0)= -2" data-latex="-{y} = {x} + {x'} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20-%206%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-4 \, {x} = -{y'} - 6 \, {y} \hspace{2em}x(0)= -5" title="-4 \, {x} = -{y'} - 6 \, {y} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} = -{y'} - 6 \, {y} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" alt="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4799" title="S2 | Linear ODE system with elimination | ver. 4799"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -3 \, {y} - 2 \, {x} \hspace{2em}x(0)= 4" alt="{x'} = -3 \, {y} - 2 \, {x} \hspace{2em}x(0)= 4" title="{x'} = -3 \, {y} - 2 \, {x} \hspace{2em}x(0)= 4" data-latex="{x'} = -3 \, {y} - 2 \, {x} \hspace{2em}x(0)= 4"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -1" alt="-7 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -1" title="-7 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -1" data-latex="-7 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20=%20-3%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%204" alt="{x'} = -3 \, {y} - 2 \, {x} \hspace{2em}x(0)= 4" title="{x'} = -3 \, {y} - 2 \, {x} \hspace{2em}x(0)= 4" data-latex="{x'} = -3 \, {y} - 2 \, {x} \hspace{2em}x(0)= 4"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7%20%5C,%20%7By%7D%20=%20%7By'%7D%20+%2012%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-7 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -1" title="-7 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -1" data-latex="-7 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" alt="x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" title="x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" data-latex="x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" alt="y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%203%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" title="x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" data-latex="x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7372" title="S2 | Linear ODE system with elimination | ver. 7372"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 3 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" alt="-{x'} + 3 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" title="-{x'} + 3 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" data-latex="-{x'} + 3 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= -1" alt="-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= -1" title="-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20+%203%20%5C,%20%7Bx%7D%20=%20-2%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-{x'} + 3 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" title="-{x'} + 3 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" data-latex="-{x'} + 3 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= -1" title="-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" alt="x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" alt="y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" title="y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" title="y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7158" title="S2 | Linear ODE system with elimination | ver. 7158"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x} = 4 \, {y} + {x'} \hspace{2em}x(0)= 3" alt="3 \, {x} = 4 \, {y} + {x'} \hspace{2em}x(0)= 3" title="3 \, {x} = 4 \, {y} + {x'} \hspace{2em}x(0)= 3" data-latex="3 \, {x} = 4 \, {y} + {x'} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 9 \, {x} = 8 \, {y} \hspace{2em}x(0)= -10" alt="{y'} + 9 \, {x} = 8 \, {y} \hspace{2em}x(0)= -10" title="{y'} + 9 \, {x} = 8 \, {y} \hspace{2em}x(0)= -10" data-latex="{y'} + 9 \, {x} = 8 \, {y} \hspace{2em}x(0)= -10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx%7D%20=%204%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="3 \, {x} = 4 \, {y} + {x'} \hspace{2em}x(0)= 3" title="3 \, {x} = 4 \, {y} + {x'} \hspace{2em}x(0)= 3" data-latex="3 \, {x} = 4 \, {y} + {x'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%209%20%5C,%20%7Bx%7D%20=%208%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="{y'} + 9 \, {x} = 8 \, {y} \hspace{2em}x(0)= -10" title="{y'} + 9 \, {x} = 8 \, {y} \hspace{2em}x(0)= -10" data-latex="{y'} + 9 \, {x} = 8 \, {y} \hspace{2em}x(0)= -10"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" alt="x= 4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="x= 4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="x= 4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" alt="y= -9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="y= -9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%204%20%5C,%20e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= 4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="x= 4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="x= 4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-9%20%5C,%20e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="y= -9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-9243" title="S2 | Linear ODE system with elimination | ver. 9243"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 4 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" alt="-{x'} + 4 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" title="-{x'} + 4 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" data-latex="-{x'} + 4 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {y} - {y'} - 2 \, {x} = 0 \hspace{2em}x(0)= -1" alt="9 \, {y} - {y'} - 2 \, {x} = 0 \hspace{2em}x(0)= -1" title="9 \, {y} - {y'} - 2 \, {x} = 0 \hspace{2em}x(0)= -1" data-latex="9 \, {y} - {y'} - 2 \, {x} = 0 \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20+%204%20%5C,%20%7Bx%7D%20=%20-2%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-{x'} + 4 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" title="-{x'} + 4 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1" data-latex="-{x'} + 4 \, {x} = -2 \, {y} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9%20%5C,%20%7By%7D%20-%20%7By'%7D%20-%202%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="9 \, {y} - {y'} - 2 \, {x} = 0 \hspace{2em}x(0)= -1" title="9 \, {y} - {y'} - 2 \, {x} = 0 \hspace{2em}x(0)= -1" data-latex="9 \, {y} - {y'} - 2 \, {x} = 0 \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" alt="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-2832" title="S2 | Linear ODE system with elimination | ver. 2832"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {y} + {x'} = -{x} \hspace{2em}x(0)= 5" alt="-4 \, {y} + {x'} = -{x} \hspace{2em}x(0)= 5" title="-4 \, {y} + {x'} = -{x} \hspace{2em}x(0)= 5" data-latex="-4 \, {y} + {x'} = -{x} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y} = -{y'} - {x} \hspace{2em}x(0)= -2" alt="6 \, {y} = -{y'} - {x} \hspace{2em}x(0)= -2" title="6 \, {y} = -{y'} - {x} \hspace{2em}x(0)= -2" data-latex="6 \, {y} = -{y'} - {x} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20=%20-%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-4 \, {y} + {x'} = -{x} \hspace{2em}x(0)= 5" title="-4 \, {y} + {x'} = -{x} \hspace{2em}x(0)= 5" data-latex="-4 \, {y} + {x'} = -{x} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By%7D%20=%20-%7By'%7D%20-%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="6 \, {y} = -{y'} - {x} \hspace{2em}x(0)= -2" title="6 \, {y} = -{y'} - {x} \hspace{2em}x(0)= -2" data-latex="6 \, {y} = -{y'} - {x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 4 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" alt="x= 4 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 4 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 4 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 4 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 4 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 4 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8231" title="S2 | Linear ODE system with elimination | ver. 8231"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x} = {x'} + 4 \, {y} \hspace{2em}x(0)= -5" alt="3 \, {x} = {x'} + 4 \, {y} \hspace{2em}x(0)= -5" title="3 \, {x} = {x'} + 4 \, {y} \hspace{2em}x(0)= -5" data-latex="3 \, {x} = {x'} + 4 \, {y} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y} + {y'} = -9 \, {x} \hspace{2em}x(0)= 8" alt="-8 \, {y} + {y'} = -9 \, {x} \hspace{2em}x(0)= 8" title="-8 \, {y} + {y'} = -9 \, {x} \hspace{2em}x(0)= 8" data-latex="-8 \, {y} + {y'} = -9 \, {x} \hspace{2em}x(0)= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="3 \, {x} = {x'} + 4 \, {y} \hspace{2em}x(0)= -5" title="3 \, {x} = {x'} + 4 \, {y} \hspace{2em}x(0)= -5" data-latex="3 \, {x} = {x'} + 4 \, {y} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%20-9%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%208" alt="-8 \, {y} + {y'} = -9 \, {x} \hspace{2em}x(0)= 8" title="-8 \, {y} + {y'} = -9 \, {x} \hspace{2em}x(0)= 8" data-latex="-8 \, {y} + {y'} = -9 \, {x} \hspace{2em}x(0)= 8"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" alt="x= -4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="x= -4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="x= -4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" alt="y= 9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="y= 9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="y= 9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= -4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="x= -4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="x= -4 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%209%20%5C,%20e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= 9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" title="y= 9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}" data-latex="y= 9 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7588" title="S2 | Linear ODE system with elimination | ver. 7588"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} = -{x'} + 4 \, {y} \hspace{2em}x(0)= -3" alt="2 \, {x} = -{x'} + 4 \, {y} \hspace{2em}x(0)= -3" title="2 \, {x} = -{x'} + 4 \, {y} \hspace{2em}x(0)= -3" data-latex="2 \, {x} = -{x'} + 4 \, {y} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y} = {y'} - {x} \hspace{2em}x(0)= -2" alt="-5 \, {y} = {y'} - {x} \hspace{2em}x(0)= -2" title="-5 \, {y} = {y'} - {x} \hspace{2em}x(0)= -2" data-latex="-5 \, {y} = {y'} - {x} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="2 \, {x} = -{x'} + 4 \, {y} \hspace{2em}x(0)= -3" title="2 \, {x} = -{x'} + 4 \, {y} \hspace{2em}x(0)= -3" data-latex="2 \, {x} = -{x'} + 4 \, {y} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20%7By%7D%20=%20%7By'%7D%20-%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-5 \, {y} = {y'} - {x} \hspace{2em}x(0)= -2" title="-5 \, {y} = {y'} - {x} \hspace{2em}x(0)= -2" data-latex="-5 \, {y} = {y'} - {x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" alt="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" alt="y= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3863" title="S2 | Linear ODE system with elimination | ver. 3863"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -5" alt="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -5" title="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -5" data-latex="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} - 2 \, {y} = {x} \hspace{2em}x(0)= 0" alt="{y'} - 2 \, {y} = {x} \hspace{2em}x(0)= 0" title="{y'} - 2 \, {y} = {x} \hspace{2em}x(0)= 0" data-latex="{y'} - 2 \, {y} = {x} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx%7D%20-%20%7Bx'%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -5" title="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -5" data-latex="0 = -{x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20-%202%20%5C,%20%7By%7D%20=%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{y'} - 2 \, {y} = {x} \hspace{2em}x(0)= 0" title="{y'} - 2 \, {y} = {x} \hspace{2em}x(0)= 0" data-latex="{y'} - 2 \, {y} = {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" alt="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3393" title="S2 | Linear ODE system with elimination | ver. 3393"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -1" alt="-6 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -1" title="-6 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -1" data-latex="-6 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 5" alt="6 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 5" title="6 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 5" data-latex="6 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-6 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -1" title="-6 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -1" data-latex="-6 \, {y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7Bx%7D%20=%20-%7By%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="6 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 5" title="6 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 5" data-latex="6 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" alt="x= -3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="x= -3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="x= -3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-3%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="x= -3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="x= -3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0645" title="S2 | Linear ODE system with elimination | ver. 0645"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} + {x'} - {x} = 0 \hspace{2em}x(0)= 5" alt="-6 \, {y} + {x'} - {x} = 0 \hspace{2em}x(0)= 5" title="-6 \, {y} + {x'} - {x} = 0 \hspace{2em}x(0)= 5" data-latex="-6 \, {y} + {x'} - {x} = 0 \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 6 \, {y} - {y'} + 6 \, {x} \hspace{2em}x(0)= 1" alt="0 = 6 \, {y} - {y'} + 6 \, {x} \hspace{2em}x(0)= 1" title="0 = 6 \, {y} - {y'} + 6 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = 6 \, {y} - {y'} + 6 \, {x} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20-%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%205" alt="-6 \, {y} + {x'} - {x} = 0 \hspace{2em}x(0)= 5" title="-6 \, {y} + {x'} - {x} = 0 \hspace{2em}x(0)= 5" data-latex="-6 \, {y} + {x'} - {x} = 0 \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%206%20%5C,%20%7By%7D%20-%20%7By'%7D%20+%206%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = 6 \, {y} - {y'} + 6 \, {x} \hspace{2em}x(0)= 1" title="0 = 6 \, {y} - {y'} + 6 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = 6 \, {y} - {y'} + 6 \, {x} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" alt="x= 2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" alt="y= 3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="y= 3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="y= 3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="y= 3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="y= 3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1500" title="S2 | Linear ODE system with elimination | ver. 1500"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 0" alt="{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 0" title="{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 0" data-latex="{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5" alt="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5" title="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5" data-latex="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 0" title="{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 0" data-latex="{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5" title="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5" data-latex="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="x= e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" alt="y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4446" title="S2 | Linear ODE system with elimination | ver. 4446"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} = {x'} - 5 \, {x} \hspace{2em}x(0)= -3" alt="4 \, {y} = {x'} - 5 \, {x} \hspace{2em}x(0)= -3" title="4 \, {y} = {x'} - 5 \, {x} \hspace{2em}x(0)= -3" data-latex="4 \, {y} = {x'} - 5 \, {x} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + 2 \, {y} = -{x} \hspace{2em}x(0)= -2" alt="-{y'} + 2 \, {y} = -{x} \hspace{2em}x(0)= -2" title="-{y'} + 2 \, {y} = -{x} \hspace{2em}x(0)= -2" data-latex="-{y'} + 2 \, {y} = -{x} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20-%205%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="4 \, {y} = {x'} - 5 \, {x} \hspace{2em}x(0)= -3" title="4 \, {y} = {x'} - 5 \, {x} \hspace{2em}x(0)= -3" data-latex="4 \, {y} = {x'} - 5 \, {x} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%202%20%5C,%20%7By%7D%20=%20-%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y'} + 2 \, {y} = -{x} \hspace{2em}x(0)= -2" title="-{y'} + 2 \, {y} = -{x} \hspace{2em}x(0)= -2" data-latex="-{y'} + 2 \, {y} = -{x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(6 \, t\right)} + e^{t}" alt="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" title="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -4 \, e^{\left(6 \, t\right)} + e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(6 \, t\right)} - e^{t}" alt="y= -e^{\left(6 \, t\right)} - e^{t}" title="y= -e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -e^{\left(6 \, t\right)} - e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" title="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -4 \, e^{\left(6 \, t\right)} + e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="y= -e^{\left(6 \, t\right)} - e^{t}" title="y= -e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -e^{\left(6 \, t\right)} - e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5204" title="S2 | Linear ODE system with elimination | ver. 5204"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -2" alt="-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -2" title="-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -2" data-latex="-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} + 12 \, {x} = {y'} \hspace{2em}x(0)= 7" alt="{y} + 12 \, {x} = {y'} \hspace{2em}x(0)= 7" title="{y} + 12 \, {x} = {y'} \hspace{2em}x(0)= 7" data-latex="{y} + 12 \, {x} = {y'} \hspace{2em}x(0)= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-3%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -2" title="-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -2" data-latex="-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20+%2012%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="{y} + 12 \, {x} = {y'} \hspace{2em}x(0)= 7" title="{y} + 12 \, {x} = {y'} \hspace{2em}x(0)= 7" data-latex="{y} + 12 \, {x} = {y'} \hspace{2em}x(0)= 7"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" alt="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" alt="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1758" title="S2 | Linear ODE system with elimination | ver. 1758"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {x'} - {y} \hspace{2em}x(0)= 0" alt="{x} = {x'} - {y} \hspace{2em}x(0)= 0" title="{x} = {x'} - {y} \hspace{2em}x(0)= 0" data-latex="{x} = {x'} - {y} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} = {y'} + 2 \, {y} \hspace{2em}x(0)= 5" alt="4 \, {x} = {y'} + 2 \, {y} \hspace{2em}x(0)= 5" title="4 \, {x} = {y'} + 2 \, {y} \hspace{2em}x(0)= 5" data-latex="4 \, {x} = {y'} + 2 \, {y} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7Bx'%7D%20-%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{x} = {x'} - {y} \hspace{2em}x(0)= 0" title="{x} = {x'} - {y} \hspace{2em}x(0)= 0" data-latex="{x} = {x'} - {y} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20+%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="4 \, {x} = {y'} + 2 \, {y} \hspace{2em}x(0)= 5" title="4 \, {x} = {y'} + 2 \, {y} \hspace{2em}x(0)= 5" data-latex="4 \, {x} = {y'} + 2 \, {y} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7651" title="S2 | Linear ODE system with elimination | ver. 7651"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x} - 4 \, {y} - {x'} = 0 \hspace{2em}x(0)= 3" alt="-3 \, {x} - 4 \, {y} - {x'} = 0 \hspace{2em}x(0)= 3" title="-3 \, {x} - 4 \, {y} - {x'} = 0 \hspace{2em}x(0)= 3" data-latex="-3 \, {x} - 4 \, {y} - {x'} = 0 \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 0" alt="-{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 0" title="-{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 0" data-latex="-{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%203" alt="-3 \, {x} - 4 \, {y} - {x'} = 0 \hspace{2em}x(0)= 3" title="-3 \, {x} - 4 \, {y} - {x'} = 0 \hspace{2em}x(0)= 3" data-latex="-3 \, {x} - 4 \, {y} - {x'} = 0 \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%20-%7Bx%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 0" title="-{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 0" data-latex="-{y'} = -{x} - 2 \, {y} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 4 \, e^{\left(-2 \, t\right)} - e^{t}" alt="x= 4 \, e^{\left(-2 \, t\right)} - e^{t}" title="x= 4 \, e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= 4 \, e^{\left(-2 \, t\right)} - e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-2 \, t\right)} + e^{t}" alt="y= -e^{\left(-2 \, t\right)} + e^{t}" title="y= -e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= -e^{\left(-2 \, t\right)} + e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="x= 4 \, e^{\left(-2 \, t\right)} - e^{t}" title="x= 4 \, e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= 4 \, e^{\left(-2 \, t\right)} - e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= -e^{\left(-2 \, t\right)} + e^{t}" title="y= -e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= -e^{\left(-2 \, t\right)} + e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0263" title="S2 | Linear ODE system with elimination | ver. 0263"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} + {x'} = 2 \, {x} \hspace{2em}x(0)= 0" alt="-{y} + {x'} = 2 \, {x} \hspace{2em}x(0)= 0" title="-{y} + {x'} = 2 \, {x} \hspace{2em}x(0)= 0" data-latex="-{y} + {x'} = 2 \, {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = 5 \, {y} - {y'} \hspace{2em}x(0)= -5" alt="-4 \, {x} = 5 \, {y} - {y'} \hspace{2em}x(0)= -5" title="-4 \, {x} = 5 \, {y} - {y'} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} = 5 \, {y} - {y'} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By%7D%20+%20%7Bx'%7D%20=%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y} + {x'} = 2 \, {x} \hspace{2em}x(0)= 0" title="-{y} + {x'} = 2 \, {x} \hspace{2em}x(0)= 0" data-latex="-{y} + {x'} = 2 \, {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%205%20%5C,%20%7By%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-4 \, {x} = 5 \, {y} - {y'} \hspace{2em}x(0)= -5" title="-4 \, {x} = 5 \, {y} - {y'} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} = 5 \, {y} - {y'} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(6 \, t\right)} + e^{t}" alt="x= -e^{\left(6 \, t\right)} + e^{t}" title="x= -e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -e^{\left(6 \, t\right)} + e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(6 \, t\right)} - e^{t}" alt="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" title="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -4 \, e^{\left(6 \, t\right)} - e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="x= -e^{\left(6 \, t\right)} + e^{t}" title="x= -e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -e^{\left(6 \, t\right)} + e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" title="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -4 \, e^{\left(6 \, t\right)} - e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3474" title="S2 | Linear ODE system with elimination | ver. 3474"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -5 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" alt="-{x'} = -5 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" title="-{x'} = -5 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-{x'} = -5 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -2" alt="{x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -2" title="{x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -2" data-latex="{x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-5%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{x'} = -5 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" title="-{x'} = -5 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-{x'} = -5 \, {x} - 4 \, {y} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7By'%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -2" title="{x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -2" data-latex="{x} = {y'} - 2 \, {y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(6 \, t\right)} + e^{t}" alt="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" title="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -4 \, e^{\left(6 \, t\right)} + e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(6 \, t\right)} - e^{t}" alt="y= -e^{\left(6 \, t\right)} - e^{t}" title="y= -e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -e^{\left(6 \, t\right)} - e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" title="x= -4 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -4 \, e^{\left(6 \, t\right)} + e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="y= -e^{\left(6 \, t\right)} - e^{t}" title="y= -e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -e^{\left(6 \, t\right)} - e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4163" title="S2 | Linear ODE system with elimination | ver. 4163"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} - {x'} = 3 \, {x} \hspace{2em}x(0)= 2" alt="-{y} - {x'} = 3 \, {x} \hspace{2em}x(0)= 2" title="-{y} - {x'} = 3 \, {x} \hspace{2em}x(0)= 2" data-latex="-{y} - {x'} = 3 \, {x} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 5" alt="-4 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 5" title="-4 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 5" data-latex="-4 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By%7D%20-%20%7Bx'%7D%20=%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="-{y} - {x'} = 3 \, {x} \hspace{2em}x(0)= 2" title="-{y} - {x'} = 3 \, {x} \hspace{2em}x(0)= 2" data-latex="-{y} - {x'} = 3 \, {x} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20=%20-8%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-4 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 5" title="-4 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 5" data-latex="-4 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" alt="x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" alt="y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5235" title="S2 | Linear ODE system with elimination | ver. 5235"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -3" alt="2 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -3" title="2 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -3" data-latex="2 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} + 2 \, {y} = {y'} \hspace{2em}x(0)= -3" alt="-2 \, {x} + 2 \, {y} = {y'} \hspace{2em}x(0)= -3" title="-2 \, {x} + 2 \, {y} = {y'} \hspace{2em}x(0)= -3" data-latex="-2 \, {x} + 2 \, {y} = {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%203%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="2 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -3" title="2 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -3" data-latex="2 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-2 \, {x} + 2 \, {y} = {y'} \hspace{2em}x(0)= -3" title="-2 \, {x} + 2 \, {y} = {y'} \hspace{2em}x(0)= -3" data-latex="-2 \, {x} + 2 \, {y} = {y'} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -2 \, e^{\left(-2 \, t\right)} - e^{t}" alt="x= -2 \, e^{\left(-2 \, t\right)} - e^{t}" title="x= -2 \, e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= -2 \, e^{\left(-2 \, t\right)} - e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-2 \, t\right)} - 2 \, e^{t}" alt="y= -e^{\left(-2 \, t\right)} - 2 \, e^{t}" title="y= -e^{\left(-2 \, t\right)} - 2 \, e^{t}" data-latex="y= -e^{\left(-2 \, t\right)} - 2 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="x= -2 \, e^{\left(-2 \, t\right)} - e^{t}" title="x= -2 \, e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= -2 \, e^{\left(-2 \, t\right)} - e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7Bt%7D" alt="y= -e^{\left(-2 \, t\right)} - 2 \, e^{t}" title="y= -e^{\left(-2 \, t\right)} - 2 \, e^{t}" data-latex="y= -e^{\left(-2 \, t\right)} - 2 \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7940" title="S2 | Linear ODE system with elimination | ver. 7940"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= -2" alt="3 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= -2" title="3 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= -2" data-latex="3 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8 \, {y} = -12 \, {x} - {y'} \hspace{2em}x(0)= 7" alt="8 \, {y} = -12 \, {x} - {y'} \hspace{2em}x(0)= 7" title="8 \, {y} = -12 \, {x} - {y'} \hspace{2em}x(0)= 7" data-latex="8 \, {y} = -12 \, {x} - {y'} \hspace{2em}x(0)= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%20-3%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="3 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= -2" title="3 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= -2" data-latex="3 \, {x} + {x'} = -3 \, {y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8%20%5C,%20%7By%7D%20=%20-12%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="8 \, {y} = -12 \, {x} - {y'} \hspace{2em}x(0)= 7" title="8 \, {y} = -12 \, {x} - {y'} \hspace{2em}x(0)= 7" data-latex="8 \, {y} = -12 \, {x} - {y'} \hspace{2em}x(0)= 7"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(-12 \, t\right)} - 3 \, e^{t}" alt="x= e^{\left(-12 \, t\right)} - 3 \, e^{t}" title="x= e^{\left(-12 \, t\right)} - 3 \, e^{t}" data-latex="x= e^{\left(-12 \, t\right)} - 3 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 3 \, e^{\left(-12 \, t\right)} + 4 \, e^{t}" alt="y= 3 \, e^{\left(-12 \, t\right)} + 4 \, e^{t}" title="y= 3 \, e^{\left(-12 \, t\right)} + 4 \, e^{t}" data-latex="y= 3 \, e^{\left(-12 \, t\right)} + 4 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7Bt%7D" alt="x= e^{\left(-12 \, t\right)} - 3 \, e^{t}" title="x= e^{\left(-12 \, t\right)} - 3 \, e^{t}" data-latex="x= e^{\left(-12 \, t\right)} - 3 \, e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%203%20%5C,%20e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7Bt%7D" alt="y= 3 \, e^{\left(-12 \, t\right)} + 4 \, e^{t}" title="y= 3 \, e^{\left(-12 \, t\right)} + 4 \, e^{t}" data-latex="y= 3 \, e^{\left(-12 \, t\right)} + 4 \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7390" title="S2 | Linear ODE system with elimination | ver. 7390"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -{y} - {x} \hspace{2em}x(0)= 0" alt="{x'} = -{y} - {x} \hspace{2em}x(0)= 0" title="{x'} = -{y} - {x} \hspace{2em}x(0)= 0" data-latex="{x'} = -{y} - {x} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} = -4 \, {x} - {y'} \hspace{2em}x(0)= -5" alt="-2 \, {y} = -4 \, {x} - {y'} \hspace{2em}x(0)= -5" title="-2 \, {y} = -4 \, {x} - {y'} \hspace{2em}x(0)= -5" data-latex="-2 \, {y} = -4 \, {x} - {y'} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20=%20-%7By%7D%20-%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{x'} = -{y} - {x} \hspace{2em}x(0)= 0" title="{x'} = -{y} - {x} \hspace{2em}x(0)= 0" data-latex="{x'} = -{y} - {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-2 \, {y} = -4 \, {x} - {y'} \hspace{2em}x(0)= -5" title="-2 \, {y} = -4 \, {x} - {y'} \hspace{2em}x(0)= -5" data-latex="-2 \, {y} = -4 \, {x} - {y'} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6154" title="S2 | Linear ODE system with elimination | ver. 6154"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x'} - 4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 5" alt="0 = {x'} - 4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 5" title="0 = {x'} - 4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = {x'} - 4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 2" alt="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 2" title="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 2" data-latex="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx'%7D%20-%204%20%5C,%20%7By%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="0 = {x'} - 4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 5" title="0 = {x'} - 4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = {x'} - 4 \, {y} - 4 \, {x} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-9%20%5C,%20%7By%7D%20+%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 2" title="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 2" data-latex="0 = -9 \, {y} + {x} + {y'} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" alt="x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" title="x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" alt="y= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" title="x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8775" title="S2 | Linear ODE system with elimination | ver. 8775"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 7" alt="{x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 7" title="{x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 7" data-latex="{x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 3 \, {x} + 6 \, {y} = 0 \hspace{2em}x(0)= 2" alt="{y'} + 3 \, {x} + 6 \, {y} = 0 \hspace{2em}x(0)= 2" title="{y'} + 3 \, {x} + 6 \, {y} = 0 \hspace{2em}x(0)= 2" data-latex="{y'} + 3 \, {x} + 6 \, {y} = 0 \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-%7Bx'%7D%20-%2012%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="{x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 7" title="{x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 7" data-latex="{x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 7"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%203%20%5C,%20%7Bx%7D%20+%206%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y'} + 3 \, {x} + 6 \, {y} = 0 \hspace{2em}x(0)= 2" title="{y'} + 3 \, {x} + 6 \, {y} = 0 \hspace{2em}x(0)= 2" data-latex="{y'} + 3 \, {x} + 6 \, {y} = 0 \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 3 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-10 \, t\right)}" alt="x= 3 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-10 \, t\right)}" title="x= 3 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-10 \, t\right)}" data-latex="x= 3 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-10 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-10 \, t\right)}" title="x= 3 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-10 \, t\right)}" data-latex="x= 3 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-2779" title="S2 | Linear ODE system with elimination | ver. 2779"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} - {x} = {x'} \hspace{2em}x(0)= -1" alt="2 \, {y} - {x} = {x'} \hspace{2em}x(0)= -1" title="2 \, {y} - {x} = {x'} \hspace{2em}x(0)= -1" data-latex="2 \, {y} - {x} = {x'} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {y} + 2 \, {x} - {y'} \hspace{2em}x(0)= 3" alt="0 = 2 \, {y} + 2 \, {x} - {y'} \hspace{2em}x(0)= 3" title="0 = 2 \, {y} + 2 \, {x} - {y'} \hspace{2em}x(0)= 3" data-latex="0 = 2 \, {y} + 2 \, {x} - {y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20-%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="2 \, {y} - {x} = {x'} \hspace{2em}x(0)= -1" title="2 \, {y} - {x} = {x'} \hspace{2em}x(0)= -1" data-latex="2 \, {y} - {x} = {x'} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7By%7D%20+%202%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = 2 \, {y} + 2 \, {x} - {y'} \hspace{2em}x(0)= 3" title="0 = 2 \, {y} + 2 \, {x} - {y'} \hspace{2em}x(0)= 3" data-latex="0 = 2 \, {y} + 2 \, {x} - {y'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(3 \, t\right)} - 2 \, e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} - 2 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - 2 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - 2 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= 2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= 2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= 2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} - 2 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - 2 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, t\right)} - 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= 2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= 2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-1745" title="S2 | Linear ODE system with elimination | ver. 1745"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} - {x'} = -2 \, {x} \hspace{2em}x(0)= -3" alt="4 \, {y} - {x'} = -2 \, {x} \hspace{2em}x(0)= -3" title="4 \, {y} - {x'} = -2 \, {x} \hspace{2em}x(0)= -3" data-latex="4 \, {y} - {x'} = -2 \, {x} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = -{x} + {y} \hspace{2em}x(0)= -2" alt="-{y'} = -{x} + {y} \hspace{2em}x(0)= -2" title="-{y'} = -{x} + {y} \hspace{2em}x(0)= -2" data-latex="-{y'} = -{x} + {y} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20=%20-2%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="4 \, {y} - {x'} = -2 \, {x} \hspace{2em}x(0)= -3" title="4 \, {y} - {x'} = -2 \, {x} \hspace{2em}x(0)= -3" data-latex="4 \, {y} - {x'} = -2 \, {x} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%20-%7Bx%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y'} = -{x} + {y} \hspace{2em}x(0)= -2" title="-{y'} = -{x} + {y} \hspace{2em}x(0)= -2" data-latex="-{y'} = -{x} + {y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="x= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="x= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="x= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="x= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="x= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6344" title="S2 | Linear ODE system with elimination | ver. 6344"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" alt="2 \, {x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="2 \, {x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="2 \, {x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" alt="-{y'} + 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" title="-{y'} + 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" data-latex="-{y'} + 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="2 \, {x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="2 \, {x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="2 \, {x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20=%207%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-{y'} + 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" title="-{y'} + 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" data-latex="-{y'} + 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 2 \, e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" alt="x= 2 \, e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" alt="y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6014" title="S2 | Linear ODE system with elimination | ver. 6014"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= 1" alt="2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= 1" title="2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= 1" data-latex="2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= 1" alt="-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= 1" title="-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= 1" data-latex="-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= 1" title="2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= 1" data-latex="2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20=%20-7%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= 1" title="-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= 1" data-latex="-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" alt="x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(6 \, t\right)} - e^{\left(3 \, t\right)}" alt="y= 2 \, e^{\left(6 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(6 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(6 \, t\right)} - e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%202%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(6 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(6 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(6 \, t\right)} - e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-2347" title="S2 | Linear ODE system with elimination | ver. 2347"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} + {x} = -{y} \hspace{2em}x(0)= -2" alt="{x'} + {x} = -{y} \hspace{2em}x(0)= -2" title="{x'} + {x} = -{y} \hspace{2em}x(0)= -2" data-latex="{x'} + {x} = -{y} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" alt="6 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="6 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="6 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20+%20%7Bx%7D%20=%20-%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x'} + {x} = -{y} \hspace{2em}x(0)= -2" title="{x'} + {x} = -{y} \hspace{2em}x(0)= -2" data-latex="{x'} + {x} = -{y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="6 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="6 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="6 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" alt="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-5474" title="S2 | Linear ODE system with elimination | ver. 5474"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x} = 4 \, {y} - {x'} \hspace{2em}x(0)= 5" alt="-3 \, {x} = 4 \, {y} - {x'} \hspace{2em}x(0)= 5" title="-3 \, {x} = 4 \, {y} - {x'} \hspace{2em}x(0)= 5" data-latex="-3 \, {x} = 4 \, {y} - {x'} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} - {x} - {y'} = 0 \hspace{2em}x(0)= -2" alt="-2 \, {y} - {x} - {y'} = 0 \hspace{2em}x(0)= -2" title="-2 \, {y} - {x} - {y'} = 0 \hspace{2em}x(0)= -2" data-latex="-2 \, {y} - {x} - {y'} = 0 \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx%7D%20=%204%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-3 \, {x} = 4 \, {y} - {x'} \hspace{2em}x(0)= 5" title="-3 \, {x} = 4 \, {y} - {x'} \hspace{2em}x(0)= 5" data-latex="-3 \, {x} = 4 \, {y} - {x'} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20-%20%7Bx%7D%20-%20%7By'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-2 \, {y} - {x} - {y'} = 0 \hspace{2em}x(0)= -2" title="-2 \, {y} - {x} - {y'} = 0 \hspace{2em}x(0)= -2" data-latex="-2 \, {y} - {x} - {y'} = 0 \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" alt="x= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" alt="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-8348" title="S2 | Linear ODE system with elimination | ver. 8348"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -2 \, {x} - 9 \, {y} \hspace{2em}x(0)= -10" alt="-{x'} = -2 \, {x} - 9 \, {y} \hspace{2em}x(0)= -10" title="-{x'} = -2 \, {x} - 9 \, {y} \hspace{2em}x(0)= -10" data-latex="-{x'} = -2 \, {x} - 9 \, {y} \hspace{2em}x(0)= -10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 3" alt="4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 3" title="4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 3" data-latex="4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-2%20%5C,%20%7Bx%7D%20-%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="-{x'} = -2 \, {x} - 9 \, {y} \hspace{2em}x(0)= -10" title="-{x'} = -2 \, {x} - 9 \, {y} \hspace{2em}x(0)= -10" data-latex="-{x'} = -2 \, {x} - 9 \, {y} \hspace{2em}x(0)= -10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20=%20-7%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 3" title="4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 3" data-latex="4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(11 \, t\right)} - 9 \, e^{\left(-2 \, t\right)}" alt="x= -e^{\left(11 \, t\right)} - 9 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(11 \, t\right)} - 9 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(11 \, t\right)} - 9 \, e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" alt="y= -e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="y= -e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(11 \, t\right)} - 9 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(11 \, t\right)} - 9 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(11 \, t\right)} - 9 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="y= -e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-2097" title="S2 | Linear ODE system with elimination | ver. 2097"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} + {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5" alt="4 \, {y} + {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5" title="4 \, {y} + {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5" data-latex="4 \, {y} + {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} = {y} + {y'} \hspace{2em}x(0)= 0" alt="-{x} = {y} + {y'} \hspace{2em}x(0)= 0" title="-{x} = {y} + {y'} \hspace{2em}x(0)= 0" data-latex="-{x} = {y} + {y'} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20-%202%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%205" alt="4 \, {y} + {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5" title="4 \, {y} + {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5" data-latex="4 \, {y} + {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20=%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x} = {y} + {y'} \hspace{2em}x(0)= 0" title="-{x} = {y} + {y'} \hspace{2em}x(0)= 0" data-latex="-{x} = {y} + {y'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="x= 4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="x= 4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="x= 4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= 4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="x= 4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="x= 4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-7727" title="S2 | Linear ODE system with elimination | ver. 7727"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x} + {x'} = 3 \, {y} \hspace{2em}x(0)= -2" alt="-3 \, {x} + {x'} = 3 \, {y} \hspace{2em}x(0)= -2" title="-3 \, {x} + {x'} = 3 \, {y} \hspace{2em}x(0)= -2" data-latex="-3 \, {x} + {x'} = 3 \, {y} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -7" alt="12 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -7" title="12 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -7" data-latex="12 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%203%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-3 \, {x} + {x'} = 3 \, {y} \hspace{2em}x(0)= -2" title="-3 \, {x} + {x'} = 3 \, {y} \hspace{2em}x(0)= -2" data-latex="-3 \, {x} + {x'} = 3 \, {y} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7By%7D%20-%20%7By'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-7" alt="12 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -7" title="12 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -7" data-latex="12 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -7"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -3 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}" alt="x= -3 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}" title="x= -3 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -3 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" alt="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-3%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}" title="x= -3 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -3 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-0036" title="S2 | Linear ODE system with elimination | ver. 0036"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -5" alt="-4 \, {x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -5" title="-4 \, {x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - 9 \, {y} = -{x} \hspace{2em}x(0)= -2" alt="-{y'} - 9 \, {y} = -{x} \hspace{2em}x(0)= -2" title="-{y'} - 9 \, {y} = -{x} \hspace{2em}x(0)= -2" data-latex="-{y'} - 9 \, {y} = -{x} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-4 \, {x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -5" title="-4 \, {x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -5" data-latex="-4 \, {x} - 4 \, {y} = {x'} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20-%209%20%5C,%20%7By%7D%20=%20-%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y'} - 9 \, {y} = -{x} \hspace{2em}x(0)= -2" title="-{y'} - 9 \, {y} = -{x} \hspace{2em}x(0)= -2" data-latex="-{y'} - 9 \, {y} = -{x} \hspace{2em}x(0)= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="x= -4 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="y= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6681" title="S2 | Linear ODE system with elimination | ver. 6681"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= 2" alt="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= 2" title="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= 2" data-latex="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= -5" alt="-3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= -5" title="-3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= -5" data-latex="-3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20+%202%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= 2" title="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= 2" data-latex="{y} + 2 \, {x} = -{x'} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By%7D%20-%204%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= -5" title="-3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= -5" data-latex="-3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" alt="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" alt="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4233" title="S2 | Linear ODE system with elimination | ver. 4233"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -5" alt="-4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -5" title="-4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -5" data-latex="-4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} - {x} = -{y'} \hspace{2em}x(0)= 0" alt="-2 \, {y} - {x} = -{y'} \hspace{2em}x(0)= 0" title="-2 \, {y} - {x} = -{y'} \hspace{2em}x(0)= 0" data-latex="-2 \, {y} - {x} = -{y'} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By%7D%20-%205%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -5" title="-4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -5" data-latex="-4 \, {y} - 5 \, {x} + {x'} = 0 \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20-%20%7Bx%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-2 \, {y} - {x} = -{y'} \hspace{2em}x(0)= 0" title="-2 \, {y} - {x} = -{y'} \hspace{2em}x(0)= 0" data-latex="-2 \, {y} - {x} = -{y'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -4 \, e^{\left(6 \, t\right)} - e^{t}" alt="x= -4 \, e^{\left(6 \, t\right)} - e^{t}" title="x= -4 \, e^{\left(6 \, t\right)} - e^{t}" data-latex="x= -4 \, e^{\left(6 \, t\right)} - e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(6 \, t\right)} + e^{t}" alt="y= -e^{\left(6 \, t\right)} + e^{t}" title="y= -e^{\left(6 \, t\right)} + e^{t}" data-latex="y= -e^{\left(6 \, t\right)} + e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-4%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="x= -4 \, e^{\left(6 \, t\right)} - e^{t}" title="x= -4 \, e^{\left(6 \, t\right)} - e^{t}" data-latex="x= -4 \, e^{\left(6 \, t\right)} - e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= -e^{\left(6 \, t\right)} + e^{t}" title="y= -e^{\left(6 \, t\right)} + e^{t}" data-latex="y= -e^{\left(6 \, t\right)} + e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-4975" title="S2 | Linear ODE system with elimination | ver. 4975"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y} + {x} + {x'} \hspace{2em}x(0)= 2" alt="0 = -{y} + {x} + {x'} \hspace{2em}x(0)= 2" title="0 = -{y} + {x} + {x'} \hspace{2em}x(0)= 2" data-latex="0 = -{y} + {x} + {x'} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" alt="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By%7D%20+%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="0 = -{y} + {x} + {x'} \hspace{2em}x(0)= 2" title="0 = -{y} + {x} + {x'} \hspace{2em}x(0)= 2" data-latex="0 = -{y} + {x} + {x'} \hspace{2em}x(0)= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20-%20%7By'%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" alt="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-3921" title="S2 | Linear ODE system with elimination | ver. 3921"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -4 \, {y} - 2 \, {x} \hspace{2em}x(0)= 5" alt="{x'} = -4 \, {y} - 2 \, {x} \hspace{2em}x(0)= 5" title="{x'} = -4 \, {y} - 2 \, {x} \hspace{2em}x(0)= 5" data-latex="{x'} = -4 \, {y} - 2 \, {x} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = 5 \, {y} + {x} \hspace{2em}x(0)= 0" alt="-{y'} = 5 \, {y} + {x} \hspace{2em}x(0)= 0" title="-{y'} = 5 \, {y} + {x} \hspace{2em}x(0)= 0" data-latex="-{y'} = 5 \, {y} + {x} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx'%7D%20=%20-4%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="{x'} = -4 \, {y} - 2 \, {x} \hspace{2em}x(0)= 5" title="{x'} = -4 \, {y} - 2 \, {x} \hspace{2em}x(0)= 5" data-latex="{x'} = -4 \, {y} - 2 \, {x} \hspace{2em}x(0)= 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%205%20%5C,%20%7By%7D%20+%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y'} = 5 \, {y} + {x} \hspace{2em}x(0)= 0" title="-{y'} = 5 \, {y} + {x} \hspace{2em}x(0)= 0" data-latex="-{y'} = 5 \, {y} + {x} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" alt="x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" alt="y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S2-6894" title="S2 | Linear ODE system with elimination | ver. 6894"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S2.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} - 4 \, {x} = {x'} \hspace{2em}x(0)= 0" alt="{y} - 4 \, {x} = {x'} \hspace{2em}x(0)= 0" title="{y} - 4 \, {x} = {x'} \hspace{2em}x(0)= 0" data-latex="{y} - 4 \, {x} = {x'} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - 4 \, {x} = 9 \, {y} \hspace{2em}x(0)= -3" alt="-{y'} - 4 \, {x} = 9 \, {y} \hspace{2em}x(0)= -3" title="-{y'} - 4 \, {x} = 9 \, {y} \hspace{2em}x(0)= -3" data-latex="-{y'} - 4 \, {x} = 9 \, {y} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Find the solution to the given system of IVPs.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20-%204%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{y} - 4 \, {x} = {x'} \hspace{2em}x(0)= 0" title="{y} - 4 \, {x} = {x'} \hspace{2em}x(0)= 0" data-latex="{y} - 4 \, {x} = {x'} \hspace{2em}x(0)= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20-%204%20%5C,%20%7Bx%7D%20=%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} - 4 \, {x} = 9 \, {y} \hspace{2em}x(0)= -3" title="-{y'} - 4 \, {x} = 9 \, {y} \hspace{2em}x(0)= -3" data-latex="-{y'} - 4 \, {x} = 9 \, {y} \hspace{2em}x(0)= -3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" alt="x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" alt="y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item></objectbank>
</questestinterop>
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