We cannot use Cramer's Rule because there are only two equations (the third is a repeat of the second). So I'll solve this in terms of one of the variables. Then I can illustrate Cramer's Rule by creating a third equation after giving this variable an arbitrary value.
(1) 2x-3y=80-z
(2) x-2y=46+z
Multiply (2) by -2: -2x+4y=-92-2z (3)
Add (1) and (3): y=-12-3z. From (2), x=46+2y+z=46-24-6z+z=22-5z.
So we have x=22-5z and y=-12-3z.
Let z=4. Then x=22-20=2, y=-12-12=-24.
Now we can create a third equation: 4x+y+4z=8-24+16=0.
Using 2x-3y+z=80, x-2y-z=46, 4x+y+4z=0, create and evaluate the determinant of coefficients, Δ=
| 2 -3 1 |
| 1 -2 -1 | = 2(-8+1)+3(4+4)+1(1+8) = -14+24+9=19
| 4 1 4 |
Δx=
| 80 -3 1 |
| 46 -2 -1 | = 80(-8+1)+3(184-0)+1(46-0) = -560+552+46=38
| 0 1 4 |
x=Δx/Δ=38/19=2.
Δy=
| 2 80 1 |
| 1 46 -1 | = 2(184)-80(4+4)+1(-184) = 368-640-184=-456
| 4 0 4 |
y=Δy/Δ=-456/19=-24.
Δz=
| 2 -3 80 |
| 1 -2 46 | = 2(-46)+3(-184)+80(1+8) = -92-552+720=76
| 4 1 0 |
z=Δz/Δ=76/19=4.
x=2, y=-24, z=4.