There’s a constant missing in the first line so assume 2x-3y+z=a where we can guess a later.
D=
| 2 -3 1 |
| 1 -1 2 |
| 3 1 -1 |
= (2)[(-1)(-1)-(2)(1)]-(-3)[(1)(-1)-(2)(3)]+(1)[(1)(1)-(-1)(3)]
= (2)(-1)+(3)(-7)+(1)(4) = -2-21+4 = -19, so D=-19.
To find x:
| a -3 1 |
| -3 -1 2 |
| 9 1 -1 |
= a(-1)-(-3)[(-3)(-1)-(2)(9)]+(1)[(-3)(1)-(-1)(9)]
= -a+(3)(-15)+(1)(6) = -a-45+6 = -a-39, x=(-a-39)/(-19).
To find y:
| 2 a 1 |
| 1 -3 2 |
| 3 9 -1 |
= (2)(-15)-a(-7)+(1)(9)-(-3)(3) = -30+7a+18 = 7a-12, y=(7a-12)/(-19).
To find z:
| 2 -3 a |
| 1 -1 -3 |
| 3 1 9 |
= (2)(-6)+(3)(18)+a(1+3) = -12+54+4a = 4a+42, z=(4a+42)/(-19).
So x=(-a-39)/(-19), y=(7a-12)/(-19), z=(4a+42)/(-19).
If a=-1, x=2, y=1, z=-2.
The first equation should have been 2x-3y+z=-1.