Answer
See solution
Work Step by Step
\[
T(\mathbf{x})=\left[\begin{array}{c}
x_{1}+x_{2} \\
4 x_{1}+5 x_{2}
\end{array}\right]=A\left[\begin{array}{c}
x_{1} \\
x_{2}
\end{array}\right]=\left[\begin{array}{cc}
1 & 1 \\
4 & 5
\end{array}\right]\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]
\]
So
\[
T(\mathbf{x})=\left[\begin{array}{l}
3 \\
8
\end{array}\right]
\]
row decrease the augmented matrix:
\[
\begin{array}{l}
{\left[\begin{array}{lll}
1 & 1 & 3 \\
4 & 5 & 8
\end{array}\right]} \\
{\left[\begin{array}{lll}
1 & 1 & 3 \\
0 & 1 & -4
\end{array}\right]} \\
R_{2}=R_{2}-4 R_{1} \\
{\left[\begin{array}{lll}
1 & 0 & 7 \\
0 & 1 & -4
\end{array}\right]}
\end{array}
\]$R_{2}=R_{2}-4 R_{1}$
$\mathbf{x}=\left[\begin{array}{c}7 \\ -4\end{array}\right]$
