Answer
$\{(5,-3)\}$.
Work Step by Step
The given system of equations is
$\left\{\begin{matrix}
2x& +5y &=&-5\\
x& +2y &=&-1
\end{matrix}\right.$
The augmented matrix is
$\Rightarrow \left[\begin{array}{cc|c}
2 & 5 & -5\\
1 & 2 & -1
\end{array}\right]$
Perform $R_1\leftrightarrow R_2$.
Swap row one and row two.
$\Rightarrow \left[\begin{array}{cc|c}
1 & 2 & -1 \\
2 & 5 & -5
\end{array}\right]$
Perform $R_2\rightarrow R_2-2 R_1$.
$\Rightarrow \left[\begin{array}{cc|c}
1 & 2 & -1 \\
2-2(1) & 5-2(2) & -5-2(-1)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{cc|c}
1 & 2 & -1 \\
0 & 1 & -3
\end{array}\right]$
Perform $R_1\rightarrow R_1-2R_2$.
$\Rightarrow \left[\begin{array}{cc|c}
1-2(0) & 2-2(1) & -1-2(-3) \\
0 & 1 & -3
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{cc|c}
1 & 0 & 5 \\
0 & 1 & -3
\end{array}\right]$
Use back substitution to solve the linear system.
$\Rightarrow x=5$
and
$\Rightarrow y=-3$.
The solution set is $\{(x,y)\}=\{(5,-3)\}$.
