Call the equations A, B and C in order. A+B=D: 5x-4z=38. This eliminates y; now we need to use C with A or B to eliminate y again: 3A+C=E: 10x-7z=74. E-2D=F: z=-2, so 5x=38+4z=30, making x=6. From B, y=17-2x+2z=17-12-4=1. Solution: (x,y,z)=(6,1,-2).
CHECK: A: 18-1+4=21; 12+1+4=17; 6+3+2=11. All OK.