\begin{exercise}{E3}{Solving linear systems}{0010}
\begin{exerciseStatement}
Show how to find the solution set for the following vector equation \[ x_{1} \left[\begin{array}{c}
1 \\
0 \\
1 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
6 \\
1 \\
-1 \\
2
\end{array}\right] + x_{3} \left[\begin{array}{c}
5 \\
2 \\
-4 \\
2
\end{array}\right] = \left[\begin{array}{c}
-7 \\
-1 \\
0 \\
-2
\end{array}\right] .\]
\end{exerciseStatement}
\begin{exerciseAnswer}\[\mathrm{RREF} \left[\begin{array}{ccc|c}
1 & 6 & 5 & -7 \\
0 & 1 & 2 & -1 \\
1 & -1 & -4 & 0 \\
0 & 2 & 2 & -2
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & -1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c}
-1 \\
-1 \\
0
\end{array}\right] \right\} \).
\end{exerciseAnswer}
\end{exercise}
