Answer
$x=\dfrac{1}{2}, y=2 \text{ or }\left(\dfrac{1}{2},2\right)$
Work Step by Step
We will write the system $\left\{\begin{array}{r}{6x+5y=13}\\{2x+2y=5}\end{array}\right.$ in matrix form as: $AX=B$
where, $X=\left[\begin{array}{l}x\\y \end{array}\right]$
We have: $A=\left[\begin{array}{ll}{6}&{5}\\{2}&{2}\end{array}\right]$, and its inverse is: $A^{-1}=\left[\begin{array}{rr}{1}&{-5/2}\\{-1}&{3}\end{array}\right]$
Thus, the solution of the given matrix can be expressed as:
$X=A^{-1}B=\left[\begin{array}{rr}{1}&{-5/2}\\{-1}&{3}\end{array}\right]\left[\begin{array}{l}
13\\5\end{array}\right]$
$\left[\begin{array}{l}
x\\y \end{array}\right]=\left[\begin{array}{l} 13-\dfrac{25}{2}\\
-13+15 \end{array}\right]=\left[\begin{array}{l}
\dfrac{1}{2}\\2\end{array}\right]$
So, our solution is:: $(x, y)=\left(\dfrac{1}{2},2\right)$