you need to eliminate 1 variable from 2 equations, then eliminate one from those 2 again
eliminate z's
1st and 2nd
multiply first by 2 and add to 2nd
6x-4y-2z=-22
2x+3y+2z=12 +
8x-y+0z=-10
8x-y=-10
eliminate z in last one
mulitply first by 3 and add to frist
9x-6y-3z=-33
x+y+3z=13 +
10x-5y+0z=-20
10x-5y=-20
our 2 equations are now
8x-y=-10 and
10x-5y=-20
eliminate y
multiply firs equaiton by -5 and add to 2nd
-40x+5y=50
10x-5y=-20 +
-30x+0y=30
-30x=30
divide by -30
x=-1
now sub back
8x-y=-10
8(-1)-y=-10
-8-y=-10
-y=-2
y=2
sub way back
x+y+3z=13
-1+2+3z=13
1+3z=13
3z=12
z=4
x=-1
y=2
z=4
(x,y,z)
(-1,2,4)