0011.ptx

<exercise checkit-seed="0011" checkit-slug="E3" checkit-title="Solving linear systems">
  <statement>
    <p>Show how to find the solution set for the following 
    
            system of linear equations.
            <me>\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}</me></p>
  </statement>
  <answer>
    <me>\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] </me>
    <p>The solution set is <m> \left\{ \left[\begin{array}{c}
2 \, a + 3 \\
a - 2 \\
a
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} </m>.</p>
  </answer>
</exercise>