Answer
See below.
Work Step by Step
If $z=0$, then $x=12\cdot0+1=1,y=10\cdot0-1=-1$, thus the equations are: $3\cdot1-4\cdot-1+4\cdot0=3+4+0=7\\1-(-1)-2\cdot0=1+1+0=2\\2\cdot1-3\cdot(-1)+6\cdot0=2+3+0=5$
These are all satisfied, thus we showed what we had to.