0011.qti

<item ident="E3-0011" title="E3 | Solving linear systems | ver. 0011">
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  <presentation>
    <material>
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        <div class="exercise-statement">
          <p>
            <strong>E3.</strong>
          </p>
          <p>Show how to find the solution set for the following system of linear equations. <p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\begin{matrix} -x_{1} &amp; - &amp; 3 \, x_{2} &amp; + &amp; 5 \, x_{3} &amp; = &amp; 3 \\ &amp; &amp; x_{2} &amp; - &amp; x_{3} &amp; = &amp; -2 \\ 2 \, x_{1} &amp; + &amp; 3 \, x_{2} &amp; - &amp; 7 \, x_{3} &amp; = &amp; 0 \\ x_{1} &amp; + &amp; x_{2} &amp; - &amp; 3 \, x_{3} &amp; = &amp; 1 \\ \end{matrix}" alt="\begin{matrix} -x_{1} &amp; - &amp; 3 \, x_{2} &amp; + &amp; 5 \, x_{3} &amp; = &amp; 3 \\ &amp; &amp; x_{2} &amp; - &amp; x_{3} &amp; = &amp; -2 \\ 2 \, x_{1} &amp; + &amp; 3 \, x_{2} &amp; - &amp; 7 \, x_{3} &amp; = &amp; 0 \\ x_{1} &amp; + &amp; x_{2} &amp; - &amp; 3 \, x_{3} &amp; = &amp; 1 \\ \end{matrix}" title="\begin{matrix} -x_{1} &amp; - &amp; 3 \, x_{2} &amp; + &amp; 5 \, x_{3} &amp; = &amp; 3 \\ &amp; &amp; x_{2} &amp; - &amp; x_{3} &amp; = &amp; -2 \\ 2 \, x_{1} &amp; + &amp; 3 \, x_{2} &amp; - &amp; 7 \, x_{3} &amp; = &amp; 0 \\ x_{1} &amp; + &amp; x_{2} &amp; - &amp; 3 \, x_{3} &amp; = &amp; 1 \\ \end{matrix}" data-latex="\begin{matrix} -x_{1} &amp; - &amp; 3 \, x_{2} &amp; + &amp; 5 \, x_{3} &amp; = &amp; 3 \\ &amp; &amp; x_{2} &amp; - &amp; x_{3} &amp; = &amp; -2 \\ 2 \, x_{1} &amp; + &amp; 3 \, x_{2} &amp; - &amp; 7 \, x_{3} &amp; = &amp; 0 \\ x_{1} &amp; + &amp; x_{2} &amp; - &amp; 3 \, x_{3} &amp; = &amp; 1 \\ \end{matrix}"/></p></p>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;E3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Show how to find the solution set for the following system of linear equations. &lt;/p&gt;
&lt;p style="text-align:center;"&gt;&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cbegin%7Bmatrix%7D%20-x_%7B1%7D%20&amp;amp;%20-%20&amp;amp;%203%20%5C,%20x_%7B2%7D%20&amp;amp;%20+%20&amp;amp;%205%20%5C,%20x_%7B3%7D%20&amp;amp;%20=%20&amp;amp;%203%20%5C%5C%20&amp;amp;%20&amp;amp;%20x_%7B2%7D%20&amp;amp;%20-%20&amp;amp;%20x_%7B3%7D%20&amp;amp;%20=%20&amp;amp;%20-2%20%5C%5C%202%20%5C,%20x_%7B1%7D%20&amp;amp;%20+%20&amp;amp;%203%20%5C,%20x_%7B2%7D%20&amp;amp;%20-%20&amp;amp;%207%20%5C,%20x_%7B3%7D%20&amp;amp;%20=%20&amp;amp;%200%20%5C%5C%20x_%7B1%7D%20&amp;amp;%20+%20&amp;amp;%20x_%7B2%7D%20&amp;amp;%20-%20&amp;amp;%203%20%5C,%20x_%7B3%7D%20&amp;amp;%20=%20&amp;amp;%201%20%5C%5C%20%5Cend%7Bmatrix%7D" alt="\begin{matrix} -x_{1} &amp;amp; - &amp;amp; 3 \, x_{2} &amp;amp; + &amp;amp; 5 \, x_{3} &amp;amp; = &amp;amp; 3 \\ &amp;amp; &amp;amp; x_{2} &amp;amp; - &amp;amp; x_{3} &amp;amp; = &amp;amp; -2 \\ 2 \, x_{1} &amp;amp; + &amp;amp; 3 \, x_{2} &amp;amp; - &amp;amp; 7 \, x_{3} &amp;amp; = &amp;amp; 0 \\ x_{1} &amp;amp; + &amp;amp; x_{2} &amp;amp; - &amp;amp; 3 \, x_{3} &amp;amp; = &amp;amp; 1 \\ \end{matrix}" title="\begin{matrix} -x_{1} &amp;amp; - &amp;amp; 3 \, x_{2} &amp;amp; + &amp;amp; 5 \, x_{3} &amp;amp; = &amp;amp; 3 \\ &amp;amp; &amp;amp; x_{2} &amp;amp; - &amp;amp; x_{3} &amp;amp; = &amp;amp; -2 \\ 2 \, x_{1} &amp;amp; + &amp;amp; 3 \, x_{2} &amp;amp; - &amp;amp; 7 \, x_{3} &amp;amp; = &amp;amp; 0 \\ x_{1} &amp;amp; + &amp;amp; x_{2} &amp;amp; - &amp;amp; 3 \, x_{3} &amp;amp; = &amp;amp; 1 \\ \end{matrix}" data-latex="\begin{matrix} -x_{1} &amp;amp; - &amp;amp; 3 \, x_{2} &amp;amp; + &amp;amp; 5 \, x_{3} &amp;amp; = &amp;amp; 3 \\ &amp;amp; &amp;amp; x_{2} &amp;amp; - &amp;amp; x_{3} &amp;amp; = &amp;amp; -2 \\ 2 \, x_{1} &amp;amp; + &amp;amp; 3 \, x_{2} &amp;amp; - &amp;amp; 7 \, x_{3} &amp;amp; = &amp;amp; 0 \\ x_{1} &amp;amp; + &amp;amp; x_{2} &amp;amp; - &amp;amp; 3 \, x_{3} &amp;amp; = &amp;amp; 1 \\ \end{matrix}"&gt;&lt;/p&gt;
&lt;/div&gt;

</mattext>
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        <response_label ident="answer1" rshuffle="No"/>
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  <itemfeedback ident="general_fb">
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          <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?\mathrm{RREF} \left[\begin{array}{ccc|c} -1 &amp; -3 &amp; 5 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 2 &amp; 3 &amp; -7 &amp; 0 \\ 1 &amp; 1 &amp; -3 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; -2 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" alt="\mathrm{RREF} \left[\begin{array}{ccc|c} -1 &amp; -3 &amp; 5 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 2 &amp; 3 &amp; -7 &amp; 0 \\ 1 &amp; 1 &amp; -3 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; -2 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" title="\mathrm{RREF} \left[\begin{array}{ccc|c} -1 &amp; -3 &amp; 5 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 2 &amp; 3 &amp; -7 &amp; 0 \\ 1 &amp; 1 &amp; -3 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; -2 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" data-latex="\mathrm{RREF} \left[\begin{array}{ccc|c} -1 &amp; -3 &amp; 5 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 2 &amp; 3 &amp; -7 &amp; 0 \\ 1 &amp; 1 &amp; -3 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; -2 &amp; 3 \\ 0 &amp; 1 &amp; -1 &amp; -2 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]"/>
            </p>
            <p>The solution set is <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" alt="\left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" title="\left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" data-latex="\left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}"/>.</p>
          </div>
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        <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?%5Cmathrm%7BRREF%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%20-1%20&amp;amp;%20-3%20&amp;amp;%205%20&amp;amp;%203%20%5C%5C%200%20&amp;amp;%201%20&amp;amp;%20-1%20&amp;amp;%20-2%20%5C%5C%202%20&amp;amp;%203%20&amp;amp;%20-7%20&amp;amp;%200%20%5C%5C%201%20&amp;amp;%201%20&amp;amp;%20-3%20&amp;amp;%201%20%5Cend%7Barray%7D%5Cright%5D%20=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%201%20&amp;amp;%200%20&amp;amp;%20-2%20&amp;amp;%203%20%5C%5C%200%20&amp;amp;%201%20&amp;amp;%20-1%20&amp;amp;%20-2%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20&amp;amp;%200%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20&amp;amp;%200%20%5Cend%7Barray%7D%5Cright%5D" alt="\mathrm{RREF} \left[\begin{array}{ccc|c} -1 &amp;amp; -3 &amp;amp; 5 &amp;amp; 3 \\ 0 &amp;amp; 1 &amp;amp; -1 &amp;amp; -2 \\ 2 &amp;amp; 3 &amp;amp; -7 &amp;amp; 0 \\ 1 &amp;amp; 1 &amp;amp; -3 &amp;amp; 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp;amp; 0 &amp;amp; -2 &amp;amp; 3 \\ 0 &amp;amp; 1 &amp;amp; -1 &amp;amp; -2 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]" title="\mathrm{RREF} \left[\begin{array}{ccc|c} -1 &amp;amp; -3 &amp;amp; 5 &amp;amp; 3 \\ 0 &amp;amp; 1 &amp;amp; -1 &amp;amp; -2 \\ 2 &amp;amp; 3 &amp;amp; -7 &amp;amp; 0 \\ 1 &amp;amp; 1 &amp;amp; -3 &amp;amp; 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp;amp; 0 &amp;amp; -2 &amp;amp; 3 \\ 0 &amp;amp; 1 &amp;amp; -1 &amp;amp; -2 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]" data-latex="\mathrm{RREF} \left[\begin{array}{ccc|c} -1 &amp;amp; -3 &amp;amp; 5 &amp;amp; 3 \\ 0 &amp;amp; 1 &amp;amp; -1 &amp;amp; -2 \\ 2 &amp;amp; 3 &amp;amp; -7 &amp;amp; 0 \\ 1 &amp;amp; 1 &amp;amp; -3 &amp;amp; 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp;amp; 0 &amp;amp; -2 &amp;amp; 3 \\ 0 &amp;amp; 1 &amp;amp; -1 &amp;amp; -2 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]"&gt;
  &lt;/p&gt;
  &lt;p&gt;The solution set is &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft%5C%7B%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%202%20%5C,%20a%20+%203%20%5C%5C%20a%20-%202%20%5C%5C%20a%20%5Cend%7Barray%7D%5Cright%5D%20%5Cmiddle%7C%5C,a%5Cin%5Cmathbb%7BR%7D%5Cright%5C%7D" alt="\left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" title="\left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" data-latex="\left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}"&gt;.&lt;/p&gt;
&lt;/div&gt;

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