2x-y=7z+1; -3x+y=z-4; add these equations together: -x=8z-3, x=3-8z.
y=z-4+3x=z-4+3(3-8z)=z-4+9-24z=5-23z.
Test: put z=0: x=3, y=5. 2x-y=1 and -9+5=-4. So the test case satisfies the two equations when z=0. Therefore we have evidence that x=3-8z and y=5-23z. Other values of z can be substituted to further test the solution.