equation 1 minus equation 2 to eliminate x: 3y+2z=3
equation 1 plus equation 3 to eliminate x: 5y+6z=5
now focus on these new system of equation, multiply 3y+2z=3 by 3:
9y+6z=9
subtract 5y+6z=5 from 9y+6z=9 to eliminate z: 4y=5, y=1
Plug y=1 into 3y+2z=3 to find z: z=0
Plug y=1, z=0 into any of the three equations to find x. I'll use equation 1:
-2x+2+0=0, x=1
so the solution is x=1, y=1, z=0