0003.ptx

<exercise checkit-seed="0003" checkit-slug="E1" checkit-title="Linear systems, vector equations, and augmented matrices">
  <statement>
    <p> Consider the augmented matrix </p>
    <me> \left[\begin{array}{ccc|c}
1 &amp; -1 &amp; -1 &amp; -6 \\
0 &amp; 1 &amp; -3 &amp; 4 \\
0 &amp; 0 &amp; 0 &amp; 1 \\
0 &amp; 2 &amp; -6 &amp; 7
\end{array}\right] </me>
    <ol>
      <li> Write a system of scalar equations corresponding to this augmented matrix. </li>
      <li> Write a vector equation corresponding to this augmented matrix. </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <me>\begin{matrix}
 x_{1} &amp;  -  &amp; x_{2} &amp;  -  &amp; x_{3} &amp; = &amp; -6 \\
 &amp;  &amp; x_{2} &amp;  -  &amp; 3 \, x_{3} &amp; = &amp; 4 \\
 &amp;  &amp;  &amp;  &amp;  0  &amp; = &amp; 1 \\
 &amp;  &amp; 2 \, x_{2} &amp;  -  &amp; 6 \, x_{3} &amp; = &amp; 7 \\
 \end{matrix}</me>
      </li>
      <li>
        <me> x_{1} \left[\begin{array}{c}
1 \\
0 \\
0 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
-1 \\
1 \\
0 \\
2
\end{array}\right] + x_{3} \left[\begin{array}{c}
-1 \\
-3 \\
0 \\
-6
\end{array}\right] = \left[\begin{array}{c}
-6 \\
4 \\
1 \\
7
\end{array}\right] </me>
      </li>
    </ol>
  </answer>
</exercise>