Call equations A, B, C.
B-C is 3x+3y=-18 or x+y=-6. Call this eqn D.
Eliminate z again: 5B-4A: 49x-26y=-144. Call this E.
E+26D: 75x=-300, so x=-4. From D y=-2 by substituting x=-4. Substitute x and y in A:
24-8-5z=-4, so 5z=20 and z=4.
Solution: x=-4, y=-2 and z=4. Check these values in A: 24-8-20=-4; in B: -20+4-16=-32; in C: -8+10-16=-14.