\begin{exercise}{E3}{Solving linear systems}{0006}
\begin{exerciseStatement}
Show how to find the solution set for the following system of linear equations. \[\begin{matrix}
x_{1} & & & - & x_{3} & = & 3 \\
5 \, x_{1} & + & x_{2} & & & = & 8 \\
-x_{1} & - & x_{2} & - & 3 \, x_{3} & = & 3 \\
-2 \, x_{1} & - & x_{2} & + & x_{3} & = & -3 \\
\end{matrix}\]
\end{exerciseStatement}
\begin{exerciseAnswer}\[\mathrm{RREF} \left[\begin{array}{ccc|c}
1 & 0 & -1 & 3 \\
5 & 1 & 0 & 8 \\
-1 & -1 & -3 & 3 \\
-2 & -1 & 1 & -3
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & -2 \\
0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0
\end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c}
2 \\
-2 \\
-1
\end{array}\right] \right\} \).
\end{exerciseAnswer}
\end{exercise}
