Question:
$x-y+z=3$
$2 x+y-z=2$
$-x-2 y+2 z=1$
Solution:
Using the equations we get
$D=\left|\begin{array}{rrr}1 & -1 & 1 \\ 2 & 1 & -1 \\ -1 & -2 & 2\end{array}\right|$
$\Rightarrow 1(2-2)+1(4-1)+1(-4+1)=0$
$D_{1}=\left|\begin{array}{rrr}3 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & 2\end{array}\right|$
$\Rightarrow 3(2-2)+1(4+1)+1(-4-1)=0$
$D_{2}=\left|\begin{array}{rrr}1 & 3 & 1 \\ 2 & 2 & -1 \\ -1 & 1 & 2\end{array}\right|$
$\Rightarrow 1(4+1)-3(4-1)+1(2+2)=0$
$D_{3}=\left|\begin{array}{rrr}1 & -1 & 3 \\ 2 & 1 & 2 \\ -1 & -2 & 1\end{array}\right|$
$\Rightarrow 1(1+4)+1(2+2)+3(-4+1)=0$