[tex]\begin{cases}-x=-y+2\\3y+4=2x\end{cases}\implies\begin{cases}x-y=-2\\2x-3y=4\end{cases}[/tex]
In matrix form, this is
[tex]\begin{bmatrix}1&-1\\2&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-2\\4\end{bmatrix}[/tex]
The coefficient matrix has determinant
[tex]\begin{vmatrix}1&-1\\2&-3\end{vmatrix}=-3+2=-1\neq0[/tex]
so it has an inverse, which is
[tex]\begin{bmatrix}1&-1\\2&-3\end{bmatrix}^{-1}=\begin{bmatrix}3&-1\\2&-1\end{bmatrix}[/tex]
Multiply both sides by the inverse matrix:
[tex]\begin{bmatrix}3&-1\\2&-1\end{bmatrix}\begin{bmatrix}1&-1\\2&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3&-1\\2&-1\end{bmatrix}\begin{bmatrix}-2\\4\end{bmatrix}[/tex]
[tex]\implies\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-10\\-8\end{bmatrix}[/tex]
so that [tex]x=-10[/tex] and [tex]y=-8[/tex].
