Answer
$$x=\frac {18}{11},\quad y= -{\frac {19}{11}}.$$
Work Step by Step
We have the system
$$\left[\begin{array}{rrr}{2} & {-1} \\{3}& {4} \end{array}\right]\left[\begin{array}{rrr}{x}\\ {y} \end{array}\right]=\left[\begin{array}{rrr}{5} \\ {-2} \end{array}\right].$$
The coefficients matrix $A=\left[\begin{array}{rrr}{2} & {-1} \\{3}& {4} \end{array}\right]$ has the inverse
$$A^{-1}=\left[ \begin {array}{cc} \frac{4}{11}&\frac{1}{11}\\ -\frac{3}{11}&\frac{2}{11}
\end {array} \right]
.
$$
The system has the solution
$$\left[\begin{array}{rrr}{x}\\ {y} \end{array}\right]=\left[ \begin {array}{cc} \frac{4}{11}&\frac{1}{11}\\ -\frac{3}{11}&\frac{2}{11}
\end {array} \right]\left[\begin{array}{rrr}{5} \\ {-2} \end{array}\right]=\left[ \begin {array}{c} {\frac {18}{11}}\\ -{
\frac {19}{11}}\end {array} \right]
.$$
That is
$$x=\frac {18}{11},\quad y= -{
\frac {19}{11}}.$$
