Show how to find the solution set for the following system of linear equations.
\[\begin{matrix} x_{1} & - & x_{2} & - & x_{3} & = & -6 \\ & & x_{2} & - & 3 \, x_{3} & = & 4 \\ & & & & 0 & = & 1 \\ & & 2 \, x_{2} & - & 6 \, x_{3} & = & 7 \\ \end{matrix}\]
Answer:
\[\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -1 & -1 & -6 \\ 0 & 1 & -3 & 4 \\ 0 & 0 & 0 & 1 \\ 0 & 2 & -6 & 7 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & -4 & 0 \\ 0 & 1 & -3 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
The solution set is \( \left\{\right\} \).