Answer
$$x=6,y=-2,z=4$$
Work Step by Step
We solve the given system of equations using Cramer's Rule. To do this, we turn the system into two matrices. We then create x, y, and z matrices by replacing the values in the original matrix with corresponding columns in the answer matrix. We then find determinants to solve. Doing this, we find:
$$ M=\begin{pmatrix}1&1&1\\ 2&-1&-1\\ 1&-2&3\end{pmatrix} \\ M_2 = \begin{pmatrix}8\\ 10\\ 22\end{pmatrix} $$
Thus:
$$M_x=\begin{pmatrix}8&1&1\\ 10&-1&-1\\ 22&-2&3\end{pmatrix} \\ M_y=\begin{pmatrix}1&8&1\\ 2&10&-1\\ 1&22&3\end{pmatrix} \\ M_z=\begin{pmatrix}1&1&8\\ 2&-1&10\\ 1&-2&22\end{pmatrix} $$
So:
$$x=\frac{D_x}{D}=\frac{-90}{-15} =6 \\ y=\frac{D_y}{D}=\frac{30}{-15} =-2 \\z=\frac{D_z}{D}=\frac{-60}{-15}=4 $$