0010.qti

<item ident="E3-0010" title="E3 | Solving linear systems | ver. 0010">
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        <div class="exercise-statement">
          <p>
            <strong>E3.</strong>
          </p>
          <p>Show how to find the solution set for the following vector equation <p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] ." alt="x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] ." title="x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] ." data-latex="x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] ."/></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 vector equation &lt;/p&gt;
&lt;p 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_%7B1%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%201%20%5C%5C%200%20%5C%5C%201%20%5C%5C%200%20%5Cend%7Barray%7D%5Cright%5D%20+%20x_%7B2%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%206%20%5C%5C%201%20%5C%5C%20-1%20%5C%5C%202%20%5Cend%7Barray%7D%5Cright%5D%20+%20x_%7B3%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%205%20%5C%5C%202%20%5C%5C%20-4%20%5C%5C%202%20%5Cend%7Barray%7D%5Cright%5D%20=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20-7%20%5C%5C%20-1%20%5C%5C%200%20%5C%5C%20-2%20%5Cend%7Barray%7D%5Cright%5D%20." alt="x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] ." title="x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] ." data-latex="x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] ."&gt;&lt;/p&gt;
&lt;/div&gt;

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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; 6 &amp; 5 &amp; -7 \\ 0 &amp; 1 &amp; 2 &amp; -1 \\ 1 &amp; -1 &amp; -4 &amp; 0 \\ 0 &amp; 2 &amp; 2 &amp; -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; 0 &amp; -1 \\ 0 &amp; 1 &amp; 0 &amp; -1 \\ 0 &amp; 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" alt="\mathrm{RREF} \left[\begin{array}{ccc|c} 1 &amp; 6 &amp; 5 &amp; -7 \\ 0 &amp; 1 &amp; 2 &amp; -1 \\ 1 &amp; -1 &amp; -4 &amp; 0 \\ 0 &amp; 2 &amp; 2 &amp; -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; 0 &amp; -1 \\ 0 &amp; 1 &amp; 0 &amp; -1 \\ 0 &amp; 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" title="\mathrm{RREF} \left[\begin{array}{ccc|c} 1 &amp; 6 &amp; 5 &amp; -7 \\ 0 &amp; 1 &amp; 2 &amp; -1 \\ 1 &amp; -1 &amp; -4 &amp; 0 \\ 0 &amp; 2 &amp; 2 &amp; -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; 0 &amp; -1 \\ 0 &amp; 1 &amp; 0 &amp; -1 \\ 0 &amp; 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" data-latex="\mathrm{RREF} \left[\begin{array}{ccc|c} 1 &amp; 6 &amp; 5 &amp; -7 \\ 0 &amp; 1 &amp; 2 &amp; -1 \\ 1 &amp; -1 &amp; -4 &amp; 0 \\ 0 &amp; 2 &amp; 2 &amp; -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp; 0 &amp; 0 &amp; -1 \\ 0 &amp; 1 &amp; 0 &amp; -1 \\ 0 &amp; 0 &amp; 1 &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} -1 \\ -1 \\ 0 \end{array}\right] \right\}" alt="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \end{array}\right] \right\}"/>.</p>
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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%201%20&amp;amp;%206%20&amp;amp;%205%20&amp;amp;%20-7%20%5C%5C%200%20&amp;amp;%201%20&amp;amp;%202%20&amp;amp;%20-1%20%5C%5C%201%20&amp;amp;%20-1%20&amp;amp;%20-4%20&amp;amp;%200%20%5C%5C%200%20&amp;amp;%202%20&amp;amp;%202%20&amp;amp;%20-2%20%5Cend%7Barray%7D%5Cright%5D%20=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D%201%20&amp;amp;%200%20&amp;amp;%200%20&amp;amp;%20-1%20%5C%5C%200%20&amp;amp;%201%20&amp;amp;%200%20&amp;amp;%20-1%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%201%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; 6 &amp;amp; 5 &amp;amp; -7 \\ 0 &amp;amp; 1 &amp;amp; 2 &amp;amp; -1 \\ 1 &amp;amp; -1 &amp;amp; -4 &amp;amp; 0 \\ 0 &amp;amp; 2 &amp;amp; 2 &amp;amp; -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp;amp; 0 &amp;amp; 0 &amp;amp; -1 \\ 0 &amp;amp; 1 &amp;amp; 0 &amp;amp; -1 \\ 0 &amp;amp; 0 &amp;amp; 1 &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; 6 &amp;amp; 5 &amp;amp; -7 \\ 0 &amp;amp; 1 &amp;amp; 2 &amp;amp; -1 \\ 1 &amp;amp; -1 &amp;amp; -4 &amp;amp; 0 \\ 0 &amp;amp; 2 &amp;amp; 2 &amp;amp; -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp;amp; 0 &amp;amp; 0 &amp;amp; -1 \\ 0 &amp;amp; 1 &amp;amp; 0 &amp;amp; -1 \\ 0 &amp;amp; 0 &amp;amp; 1 &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; 6 &amp;amp; 5 &amp;amp; -7 \\ 0 &amp;amp; 1 &amp;amp; 2 &amp;amp; -1 \\ 1 &amp;amp; -1 &amp;amp; -4 &amp;amp; 0 \\ 0 &amp;amp; 2 &amp;amp; 2 &amp;amp; -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 &amp;amp; 0 &amp;amp; 0 &amp;amp; -1 \\ 0 &amp;amp; 1 &amp;amp; 0 &amp;amp; -1 \\ 0 &amp;amp; 0 &amp;amp; 1 &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%20-1%20%5C%5C%20-1%20%5C%5C%200%20%5Cend%7Barray%7D%5Cright%5D%20%5Cright%5C%7D" alt="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \end{array}\right] \right\}"&gt;.&lt;/p&gt;
&lt;/div&gt;

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