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