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