Show how to find the solution set for the following system of linear equations.
\[\begin{matrix} x_{1} & - & 3 \, x_{2} & + & 7 \, x_{3} & - & 4 \, x_{4} & = & 4 \\ x_{1} & - & 2 \, x_{2} & + & 4 \, x_{3} & - & 3 \, x_{4} & = & 3 \\ 2 \, x_{1} & - & 4 \, x_{2} & + & 8 \, x_{3} & - & 6 \, x_{4} & = & 6 \\ \end{matrix}\]
Answer:
\[\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & -3 & 7 & -4 & 4 \\ 1 & -2 & 4 & -3 & 3 \\ 2 & -4 & 8 & -6 & 6 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -2 & -1 & 1 \\ 0 & 1 & -3 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c} 2 \, a + b + 1 \\ 3 \, a - b - 1 \\ a \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\} \).