Call the three equations A, B and C. A+B: 10x+2z=-16. Treat z as a constant so 10x=-2z-16 or 5x=-z-8 and x=-(z+8)/5. Substitute this in A: -3(z+8)/5+2y=3z-2. Multiply through by 5: -3(z+8)+10y=15z-10. Expand brackets: -3z-24+10y=15z-10, so 10y=18z+14 or 5y=9z+7, so y=(9z+7)/5. Now we have x and y in terms of z, so we can substitute both in C: -2(z+8)/5+4(9z+7)/5+z=6. Multiply through by 5: -2z-16+36z+28+5z=30. Collect the z terms together: 39z+12=30, or 13z+4=10, so z=6/13. Now from z we can find x and y. So x=-(z+8)/5=-22/13 and y=29/13. If we plug these rather ungainly values into A, B and C, we can check if they're correct.
-66/13+58/13-18/13=-26/13=-2. Correct!
-154/13-58/13 +30/13=-182/13=-14. Correct!
-44/13+116/13+6/13=78/13=6. Correct!
We combined A and B, but we could also combine 2B+C: 16x+11z=-22. A+B: 10x+2z=-16 or 5x+z=-8. So we have two equations here (call them D and E) and two variables x and z. D-11E: -39x=66, -13x=22, so x=-22/13. And we can combine 5D-16E: 80x+55z-(80x+16z)=-110+128, so 39z=18, 13z=6, so z=6/13.