\begin{exercise}{E3}{Solving linear systems}{0008}
\begin{exerciseStatement}
Show how to find the solution set for the following system of linear equations. \[\begin{matrix}
& & 4 \, x_{2} & + & 4 \, x_{3} & = & 8 \\
-x_{1} & + & 2 \, x_{2} & - & 2 \, x_{3} & = & -2 \\
-x_{1} & - & 3 \, x_{2} & + & 3 \, x_{3} & = & 8 \\
-4 \, x_{1} & + & 8 \, x_{2} & - & 8 \, x_{3} & = & -8 \\
\end{matrix}\]
\end{exerciseStatement}
\begin{exerciseAnswer}\[\mathrm{RREF} \left[\begin{array}{ccc|c}
0 & -4 & 4 & 8 \\
-1 & 2 & -2 & -2 \\
-1 & -3 & 3 & 8 \\
-4 & 8 & -8 & -8
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & -2 \\
0 & 1 & -1 & -2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c}
-2 \\
a - 2 \\
a
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} \).
\end{exerciseAnswer}
\end{exercise}
