0012.ptx

<exercise checkit-seed="0012" checkit-slug="E1" checkit-title="Linear systems, vector equations, and augmented matrices">
  <statement>
    <p> Consider the system of equations</p>
    <me>\begin{matrix}
 x &amp;  +  &amp; 2 \, y &amp;  +  &amp; z &amp; = &amp; 5 \\
 -2 \, x &amp;  -  &amp; 3 \, y &amp;  -  &amp; z &amp; = &amp; -5 \\
 x &amp;  +  &amp; y &amp;  +  &amp; z &amp; = &amp; 4 \\
 2 \, x &amp;  +  &amp; 4 \, y &amp;  &amp;  &amp; = &amp; 2 \\
 \end{matrix}</me>
    <ol>
      <li> Write an augmented matrix corresponding to this system. </li>
      <li> Write a vector equation corresponding to this system. </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <me> \left[\begin{array}{ccc|c}
1 &amp; 2 &amp; 1 &amp; 5 \\
-2 &amp; -3 &amp; -1 &amp; -5 \\
1 &amp; 1 &amp; 1 &amp; 4 \\
2 &amp; 4 &amp; 0 &amp; 2
\end{array}\right] </me>
      </li>
      <li>
        <me> x_{1} \left[\begin{array}{c}
1 \\
-2 \\
1 \\
2
\end{array}\right] + x_{2} \left[\begin{array}{c}
2 \\
-3 \\
1 \\
4
\end{array}\right] + x_{3} \left[\begin{array}{c}
1 \\
-1 \\
1 \\
0
\end{array}\right] = \left[\begin{array}{c}
5 \\
-5 \\
4 \\
2
\end{array}\right] </me>
      </li>
    </ol>
  </answer>
</exercise>