2w+3x+y-z=9, -w+x+y+2z=10, 3w+x-y-z=-2, w-x+y-z+-6

Question

2/3x - 1/4y + 5/8z = 0, 1/5x + 2/3y - 1/4z = -7, -3/5x + 4/3y - 7/8z = -5    

solve for w,x,y,z

Answer 1

1.

Call equations A, B, C, D and I guess D should be w-x+y-z=-6.

A-D: E: w+4x=15; x=(1/4)(15-w)
C+D: F: 4w-2z=-8; 2w-z=-4; z=2w+4

We now have x and z in terms of w.
From A, y=9-2w-3x+z=9-2w-(3/4)(15-w)+2w+4=7/4+3w/4; y=(1/4)(3w+7).

All the variables are in terms of w. We just need to take one equation to find w.
Using B, w=x+y+2z-10=(15-w)/4+(3w+7)/4+4w+8-10=(22+2w)/4+4w-2.

2w=11+w+8w-4; 7w=-7, so w=-1; x=4; y=1; z=2.

CHECK:

A: -2+12+1-2=9; B: 1+4+1+4=10; C: -3+4-1-2=-2; D: -1-4+1-2=-6. All OK.

2.

A: Multiply through by 24: 16x-6y+15z=0

B: Multiply through by 60: 12x+40y-15z=-420

C: Multiply through by 120: -72x+160y-105z=-600

A+B: 28x+34y=-420; D: 14x+17y=-210; 7A+C: 40x+118y=-600; E: 20x+59y=-300

10D-7E: -243y=-2100+2100, so y=0; x=-420/28=-15; 15z=240, z=16.

CHECK:

Original A: -10+10=0

Original B: -3-4=-7

Original C: 9-14=-5

All OK.