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