Eliminate one variable at a time. We have three equations so we can solve for three variables.
(5x-y+z=4)-5(x+2y-z=5)=-11y+6z=-21
-2(x+2y-z=5)+(2x+3y-3z=5)=-y-z=-5
Now using the two yz equations above to cancel out z
(-11y+6z=-21)+6(-y-z=-5)=-17y=-51
-17y=-51 divide both sides by -17
y=3, making -y-z=-5 become:
-3-z=-5, z=2, making 5x-y+z=4 become:
5x-3+2=4
5x-1=4
5x=5
x=1
So the solution to the system of equations, (x,y,z) is (1,3,2)