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