\begin{exercise}{E1}{Linear systems, vector equations, and augmented matrices}{0000}
\begin{exerciseStatement}
Consider the augmented matrix
\[ \left[\begin{array}{cccc|c}
1 & -2 & -1 & 3 & 4 \\
2 & -3 & -3 & 5 & 6 \\
0 & -3 & 3 & 3 & 6
\end{array}\right] \]
\begin{enumerate}[(a)]
\item Write a system of scalar equations corresponding to this augmented matrix.
\item Write a vector equation corresponding to this augmented matrix.
\end{enumerate}
\end{exerciseStatement}
\begin{exerciseAnswer}
\begin{enumerate}[(a)]
\item \[\begin{matrix}
x_{1} & - & 2 \, x_{2} & - & x_{3} & + & 3 \, x_{4} & = & 4 \\
2 \, x_{1} & - & 3 \, x_{2} & - & 3 \, x_{3} & + & 5 \, x_{4} & = & 6 \\
& & 3 \, x_{2} & + & 3 \, x_{3} & + & 3 \, x_{4} & = & 6 \\
\end{matrix}\]
\item \[ x_{1} \left[\begin{array}{c}
1 \\
2 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
-2 \\
-3 \\
-3
\end{array}\right] + x_{3} \left[\begin{array}{c}
-1 \\
-3 \\
3
\end{array}\right] + x_{4} \left[\begin{array}{c}
3 \\
5 \\
3
\end{array}\right] = \left[\begin{array}{c}
4 \\
6 \\
6
\end{array}\right] \]
\end{enumerate}
\end{exerciseAnswer}
\end{exercise}
