Show how to find the solution set for the following vector equation
\[ x_{1} \left[\begin{array}{c} -3 \\ 0 \\ -1 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 7 \\ -5 \\ 3 \\ 7 \end{array}\right] = \left[\begin{array}{c} 1 \\ 6 \\ 0 \\ -2 \end{array}\right] .\]
Answer:
\[\mathrm{RREF} \left[\begin{array}{ccc|c} -3 & -1 & 7 & 1 \\ 0 & 1 & -5 & 6 \\ -1 & 0 & 3 & 0 \\ -2 & -1 & 7 & -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c} -3 \\ 1 \\ -1 \end{array}\right] \right\} \).