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