Looks like the system is
x + 4y - z = -14
5x + 6y + 3z = 4
-2x + 7y + 2z = -17
or in matrix form,
Cramer's rule says that
where
is the solution for i-th variable, and
is a modified version of
with its i-th column replaced by
.
We have 4 determinants to compute. I'll show the work for det(A) using a cofactor expansion along the first row.
The modified matrices and their determinants are
Then by Cramer's rule, the solution to the system is