\begin{exercise}{E3}{Solving linear systems}{0004}
\begin{exerciseStatement}
Show how to find the solution set for the following vector equation \[ x_{1} \left[\begin{array}{c}
-3 \\
0 \\
-1 \\
-2
\end{array}\right] + x_{2} \left[\begin{array}{c}
-1 \\
1 \\
0 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
7 \\
-5 \\
3 \\
7
\end{array}\right] = \left[\begin{array}{c}
1 \\
6 \\
0 \\
-2
\end{array}\right] .\]
\end{exerciseStatement}
\begin{exerciseAnswer}\[\mathrm{RREF} \left[\begin{array}{ccc|c}
-3 & -1 & 7 & 1 \\
0 & 1 & -5 & 6 \\
-1 & 0 & 3 & 0 \\
-2 & -1 & 7 & -2
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & -3 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0
\end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c}
-3 \\
1 \\
-1
\end{array}\right] \right\} \).
\end{exerciseAnswer}
\end{exercise}
