Since x+y=-10, y=-10-x, so we can substitute for y in the other equations:
7x-5(-10-x)-6z=200, 7x+50+5x-6z=200, 12x-6z=150, 2x-z=25.
12x+6(-10-x)+z=-35, 12x-60-6x+z=-35, 6x+z=25.
Now we have two equations and two unknowns:
2x-z=25
6x+z=25
Add them together:
8x=50, x=50/8=25/4. 2x=25/2, so z=2x-25=25/2-25=-25/2.
And y=-10-x=-10-25/4=-(10+25/4)=-65/4.
SOLUTION: x=25/4, y=-65/4, z=-25/2.