\begin{exercise}{E3}{Solving linear systems}{0002}
\begin{exerciseStatement}
Show how to find the solution set for the following system of linear equations. \[\begin{matrix}
4 \, x_{1} & + & x_{2} & + & 7 \, x_{3} & = & 6 \\
-5 \, x_{1} & - & 4 \, x_{2} & - & 6 \, x_{3} & = & -2 \\
-x_{1} & + & x_{2} & - & 3 \, x_{3} & = & -4 \\
-3 \, x_{1} & & & - & 6 \, x_{3} & = & -6 \\
\end{matrix}\]
\end{exerciseStatement}
\begin{exerciseAnswer}\[\mathrm{RREF} \left[\begin{array}{ccc|c}
4 & 1 & 7 & 6 \\
-5 & -4 & -6 & -2 \\
-1 & 1 & -3 & -4 \\
-3 & 0 & -6 & -6
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 2 & 2 \\
0 & 1 & -1 & -2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c}
-2 \, a + 2 \\
a - 2 \\
a
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} \).
\end{exerciseAnswer}
\end{exercise}
