The coefficient matrix of \(\begin{eqnarray*} 5 x + 3 y &=& - 2\\ -3 x + ~~y &=& ~~4 \end{eqnarray*} \) is \(\left[\begin{array}{rr|r} 5 & 3 & -2\\ -3 & 1 & 4 \end{array}\right]\).
\(D=\left|\begin{array}{rr} 5 & 3 \\ -3 & 1 \end{array}\right|=14\qquad D_x=\left|\begin{array}{rr} -2 & 3 \\ 4 & 1 \end{array}\right|=-14 \qquad D_y=\left|\begin{array}{rr} 5 & -2 \\ -3 & 4 \end{array}\right|=14\qquad \)
\(x=\dfrac{D_x}{D}=-1 \qquad y=\dfrac{D_y}{D}=1 \)
So the solution is \((-1,1)\).