Answer
See below
Work Step by Step
If $z=1$, then $x=12\cdot1+13=1,y=10\cdot1-1=9$, thus the equations are: $3\cdot13-4\cdot9+4\cdot1=39-36+4=7\\13-9-2\cdot1=13-9-2\\2\cdot13-3\cdot9+6\cdot1=26-27+6=5$
These are all satisfied, thus we showed what we had to.