Step 1

The given equation

$4x+5y=8$

$3x-4y=3$

In matrix form

$\left[\begin{array}{cc}4& 5\\ 3& -4\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 3\end{array}\right]$ $AX=B$

Step 2

$X={A}^{-1}BA=\left[\begin{array}{cc}4& 5\\ 3& -4\end{array}\right]{A}^{-1}=\frac{1}{-16-15}\left[\begin{array}{cc}-4& -5\\ -3& 4\end{array}\right]=\frac{-1}{31}\left[\begin{array}{cc}-4& -5\\ -3& 4\end{array}\right]$

$B=\left[\begin{array}{c}8\\ 3\end{array}\right]$

Step 3

${A}^{-1}B=\frac{-1}{31}\left[\begin{array}{cc}-4& -5\\ -3& 4\end{array}\right]$$\left[\begin{array}{c}8\\ 3\end{array}\right]$

$=\frac{-1}{31}\left[\begin{array}{c}-47\\ -12\end{array}\right]$

$x=\frac{47}{31}$

$y=\frac{12}{31}$