0005.ptx

<exercise checkit-seed="0005" 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; 7 \, x_{3} &amp;  -  &amp; 4 \, x_{4} &amp; = &amp; 4 \\
 x_{1} &amp;  -  &amp; 2 \, x_{2} &amp;  +  &amp; 4 \, x_{3} &amp;  -  &amp; 3 \, x_{4} &amp; = &amp; 3 \\
 2 \, x_{1} &amp;  -  &amp; 4 \, x_{2} &amp;  +  &amp; 8 \, x_{3} &amp;  -  &amp; 6 \, x_{4} &amp; = &amp; 6 \\
 \end{matrix}</me></p>
  </statement>
  <answer>
    <me>\mathrm{RREF} \left[\begin{array}{cccc|c}
1 &amp; -3 &amp; 7 &amp; -4 &amp; 4 \\
1 &amp; -2 &amp; 4 &amp; -3 &amp; 3 \\
2 &amp; -4 &amp; 8 &amp; -6 &amp; 6
\end{array}\right]  =  \left[\begin{array}{cccc|c}
1 &amp; 0 &amp; -2 &amp; -1 &amp; 1 \\
0 &amp; 1 &amp; -3 &amp; 1 &amp; -1 \\
0 &amp; 0 &amp; 0 &amp; 0 &amp; 0
\end{array}\right] </me>
    <p>The solution set is <m> \left\{ \left[\begin{array}{c}
2 \, a + b + 1 \\
3 \, a - b - 1 \\
a \\
b
\end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\} </m>.</p>
  </answer>
</exercise>