You just asked for z, but I'll give you the determinant for all 
three unknowns:








The only answer you want is this last one:

=

Notice that the denominators are all the same and they are made
from the coefficients of the unknowns.  Notice that each numerator
determinant is just like the denominator except for 

the 1st column of the 1st unknown x,
the 2nd column of the 2nd unknown y, and 
the 3rd column of the 3rd unknown z.  

What replaces these is the column of constants  on the right of 
the equal signs, which does not appear in the denominator.

Edwin