#### Answer

$x=9, y=23 \text{ or } (9,23)$

#### Work Step by Step

We will write the system $\left\{\begin{array}{r}{3x-y=4}\\{-2x+y=5}\end{array}\right.$ in matrix form as: $AX=B$
where, $X=\left[\begin{array}{l}x\\y \end{array}\right]$
We have: $A=\left[\begin{array}{ll}{3}&{-1}\\{-2}&{1}\end{array}\right]$, and its inverse is: $A^{-1}=\left[\begin{array}{rr}{1}&{1}\\{2}&{3}\end{array}\right]$
Thus, the solution of the given matrix can be expressed as:
$X=A^{-1}B=\left[\begin{array}{rr}{1}&{1}\\{2}&{3}\end{array}\right]\left[\begin{array}{l}
4\\5\end{array}\right]$
$\left[\begin{array}{l}
x\\y \end{array}\right]=\left[\begin{array}{l} 4+5\\
8+15 \end{array}\right]=\left[\begin{array}{l}
9\\23\end{array}\right]$
So, our solution is: $(x, y)=(9,23)$