Show how to find the solution set for the following system of linear equations.
\[\begin{matrix} -x_{1} & - & 3 \, x_{2} & + & 5 \, x_{3} & = & 3 \\ & & x_{2} & - & x_{3} & = & -2 \\ 2 \, x_{1} & + & 3 \, x_{2} & - & 7 \, x_{3} & = & 0 \\ x_{1} & + & x_{2} & - & 3 \, x_{3} & = & 1 \\ \end{matrix}\]
Answer:
\[\mathrm{RREF} \left[\begin{array}{ccc|c} -1 & -3 & 5 & 3 \\ 0 & 1 & -1 & -2 \\ 2 & 3 & -7 & 0 \\ 1 & 1 & -3 & 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & -2 & 3 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c} 2 \, a + 3 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\} \).