The given system of equation is
$5 x-y+2 z=3, x+y+z=6$ and $2 x+y-z=-5$
In matrix form, if may be represented as where
$A=\left[\begin{array}{ccc}1 & 1 & 1 \\ 5 & -1 & 2 \\ 2 & 1 & -1\end{array}\right], x=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $D=\left[\begin{array}{c}6 \\ 3 \\ -5\end{array}\right]$
Thus, $A$ is a non-singular matrix, so $A^{-1}$ exist.
$\begin{aligned} & C_{11}=(-1)^{1+1}\left|\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right|=-1 \\ & C_{12}=(-1)^{1+2}\left|\begin{array}{cc}5 & 2 \\ 2 & -1\end{array}\right|=9 \\ & C_{13}=(-1)^{1+3}\left|\begin{array}{cc}5 & -1 \\ 2 & 1\end{array}\right|=7 \\ & C_{21}=(-1)^{2+1}\left|\begin{array}{cc}1 & 1 \\ 1 & -1\end{array}\right|=2\end{aligned}$
$\begin{aligned} & C_{22}=(-1)^{2+2}\left|\begin{array}{ll}1 & 1 \\ 2 & -1\end{array}\right|=-3 \\ & C_{23}=(-1)^{2+3}\left|\begin{array}{ll}1 & 1 \\ 2 & 1\end{array}\right|=1 \\ & C_{31}=(-1)^{3+1}\left|\begin{array}{cc}1 & 1 \\ -1 & 2\end{array}\right|=3 \\ & C_{32}=(-1)^{3+2}\left|\begin{array}{ll}1 & 1 \\ 5 & 2\end{array}\right|=3 \\ & C_{33}=(-1)^{3+3}\left|\begin{array}{cc}1 & 1 \\ 5 & -1\end{array}\right|=-6\end{aligned}$
Then, $\operatorname{adj}(A)=\left[\begin{array}{ccc}-1 & 9 & 7 \\ 2 & -3 & 1 \\ 3 & 3 & -6\end{array}\right]^T=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 9 & -3 & 3 \\ 7 & 1 & -6\end{array}\right]$
$(\operatorname{adj} A) \cdot D=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 9 & -3 & 3 \\ 7 & 1 & -6\end{array}\right]\left[\begin{array}{c}6 \\ 3 \\ -5\end{array}\right]$
$=\left[\begin{array}{c}-6+6-15 \\ 54-9-15 \\ 42+3+30\end{array}\right]=\left[\begin{array}{c}-15 \\ 30 \\ 75\end{array}\right]$