\begin{exercise}{E3}{Solving linear systems}{0000}
\begin{exerciseStatement}
Show how to find the solution set for the following system of linear equations. \[\begin{matrix}
x_{1} & - & 2 \, x_{2} & - & x_{3} & + & 3 \, x_{4} & = & 4 \\
2 \, x_{1} & - & 3 \, x_{2} & - & 3 \, x_{3} & + & 5 \, x_{4} & = & 6 \\
& & 3 \, x_{2} & + & 3 \, x_{3} & + & 3 \, x_{4} & = & 6 \\
\end{matrix}\]
\end{exerciseStatement}
\begin{exerciseAnswer}\[\mathrm{RREF} \left[\begin{array}{cccc|c}
1 & -2 & -1 & 3 & 4 \\
2 & -3 & -3 & 5 & 6 \\
0 & -3 & 3 & 3 & 6
\end{array}\right] = \left[\begin{array}{cccc|c}
1 & 0 & -3 & 1 & 0 \\
0 & 1 & -1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] \]
The solution set is \( \left\{ \left[\begin{array}{c}
3 \, a - b \\
a + b - 2 \\
a \\
b
\end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\} \).
\end{exerciseAnswer}
\end{exercise}
