2x -  y + 2z = -8
-x + 3y - 4z = 15
 x + 2y - 3z =  9

[ 2  -1   2 | -8]
[-1   3  -4 | 15]
[ 1   2  -3 |  9] 

Get a zero where the -1 is on the 2nd row is by
multiplying the first row by 1 and adding it
to 2 times the second row:

1[ 2  -1   2 | -8]
2[-1   3  -4 | 15]
 [ 1   2  -3 |  9]

[ 2  -1   2 | -8]
[ 0   5  -6 | 22]
[ 1   2  -3 |  9]

Get a zero where the 1 is on the 3rd row is by
multiplying the first row by 1 and adding it
to -2 times the third row: 

 1[ 2  -1   2 | -8]
  [ 0   5  -6 | 22]
-2[ 1   2  -3 |  9]

[ 2  -1   2 | -8]
[ 0   5  -6 | 22]
[ 0  -5   8 |-26]

Get a 0 where the -5 is on the third row by
multiplying the second row by 1 and adding it
to 1 times row 3:

 [ 2  -1   2 | -8]
1[ 0   5  -6 | 22]
1[ 0  -5   8 |-26]

[ 2  -1   2 | -8]
[ 0   5  -6 | 22]
[ 0   0   2 | -4]

-1,2,-2

Get a 1 where the 2 is in the first row by
dividing the first row through by 2

Get a 1 where the 5 is in the second row by
dividing the second row through by 5

Get a 1 where the 2 is in the third row by
dividing the third row through by 2

[ 1       1 |  -4]
[ 0     1   | ]
[ 0     0    1 |  -2]

Convert back to three equations:

 = 
    = 
          

Substitute z = -2 into the second equation

    = 

    = 

        = 
        = 
        = 

Finally substitute  and 
into the first equation:

 = 

 = 







So the solution is

(x, y, z) = (-1, 2, -2)

Edwin