\begin{exercise}{E1}{Linear systems, vector equations, and augmented matrices}{0011}
\begin{exerciseStatement}
Consider the augmented matrix
\[ \left[\begin{array}{ccc|c}
-1 & -3 & 5 & 3 \\
0 & 1 & -1 & -2 \\
2 & 3 & -7 & 0 \\
1 & 1 & -3 & 1
\end{array}\right] \]
\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 \[\begin{matrix}
-x_{1} & - & 3 \, x_{2} & + & 5 \, x_{3} & = & 3 \\
& & x_{2} & - & x_{3} & = & -2 \\
2 \, x_{1} & + & 3 \, x_{2} & - & 7 \, x_{3} & = & 0 \\
x_{1} & + & x_{2} & - & 3 \, x_{3} & = & 1 \\
\end{matrix}\]
\item \[ x_{1} \left[\begin{array}{c}
-1 \\
0 \\
2 \\
1
\end{array}\right] + x_{2} \left[\begin{array}{c}
-3 \\
1 \\
3 \\
1
\end{array}\right] + x_{3} \left[\begin{array}{c}
5 \\
-1 \\
-7 \\
-3
\end{array}\right] = \left[\begin{array}{c}
3 \\
-2 \\
0 \\
1
\end{array}\right] \]
\end{enumerate}
\end{exerciseAnswer}
\end{exercise}
