0007.ptx

<exercise checkit-seed="0007" 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}
4 &amp; 2 &amp; 0 &amp; -6 \\
1 &amp; 2 &amp; -5 &amp; 0 \\
5 &amp; 4 &amp; -6 &amp; -6 \\
-3 &amp; -1 &amp; -2 &amp; 5
\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}
 4 \, x &amp;  +  &amp; 2 \, y &amp;  &amp;  &amp; = &amp; -6 \\
 x &amp;  +  &amp; 2 \, y &amp;  -  &amp; 5 \, z &amp; = &amp; 0 \\
 5 \, x &amp;  +  &amp; 4 \, y &amp;  -  &amp; 6 \, z &amp; = &amp; -6 \\
 -3 \, x &amp;  -  &amp; y &amp;  -  &amp; 2 \, z &amp; = &amp; 5 \\
 \end{matrix}</me>
      </li>
      <li>
        <me> x_{1} \left[\begin{array}{c}
4 \\
1 \\
5 \\
-3
\end{array}\right] + x_{2} \left[\begin{array}{c}
2 \\
2 \\
4 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
0 \\
-5 \\
-6 \\
-2
\end{array}\right] = \left[\begin{array}{c}
-6 \\
0 \\
-6 \\
5
\end{array}\right] </me>
      </li>
    </ol>
  </answer>
</exercise>