\begin{exercise}{E1}{Linear systems, vector equations, and augmented matrices}{0008}
\begin{exerciseStatement}
Consider the vector equation.
\[ x_{1} \left[\begin{array}{c}
0 \\
-1 \\
-1 \\
-4
\end{array}\right] + x_{2} \left[\begin{array}{c}
-4 \\
2 \\
-3 \\
8
\end{array}\right] + x_{3} \left[\begin{array}{c}
4 \\
-2 \\
3 \\
-8
\end{array}\right] = \left[\begin{array}{c}
8 \\
-2 \\
8 \\
-8
\end{array}\right] \]
\begin{enumerate}[(a)]
\item Write a system of scalar equations corresponding to this vector equation.
\item Write an augmented matrix corresponding to this vector equation.
\end{enumerate}
\end{exerciseStatement}
\begin{exerciseAnswer}
\begin{enumerate}[(a)]
\item \[\begin{matrix}
& & 4 \, x_{2} & + & 4 \, x_{3} & = & 8 \\
-x_{1} & + & 2 \, x_{2} & - & 2 \, x_{3} & = & -2 \\
-x_{1} & - & 3 \, x_{2} & + & 3 \, x_{3} & = & 8 \\
-4 \, x_{1} & + & 8 \, x_{2} & - & 8 \, x_{3} & = & -8 \\
\end{matrix}\]
\item \[ \left[\begin{array}{ccc|c}
0 & -4 & 4 & 8 \\
-1 & 2 & -2 & -2 \\
-1 & -3 & 3 & 8 \\
-4 & 8 & -8 & -8
\end{array}\right] \]
\end{enumerate}
\end{exerciseAnswer}
\end{exercise}
