\begin{exercise}{E3}{Solving linear systems}{0009}
\begin{exerciseStatement}
Show how to find the solution set for the following system of linear equations. \[\begin{matrix}
x_{1} & & & + & 3 \, x_{3} & = & -4 \\
& & x_{2} & + & 3 \, x_{3} & = & -5 \\
x_{1} & & & + & 4 \, x_{3} & = & -6 \\
3 \, x_{1} & - & 2 \, x_{2} & + & 6 \, x_{3} & = & -8 \\
\end{matrix}\]
\end{exerciseStatement}
\begin{exerciseAnswer}\[\mathrm{RREF} \left[\begin{array}{ccc|c}
1 & 0 & 3 & -4 \\
0 & 1 & 3 & -5 \\
1 & 0 & 4 & -6 \\
3 & -2 & 6 & -8
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & -2 \\
0 & 0 & 0 & 0
\end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c}
2 \\
1 \\
-2
\end{array}\right] \right\} \).
\end{exerciseAnswer}
\end{exercise}
