x-3y+2z=-21
y-2z=12
z=2
Convert to matrix form:
1 -3 2 -21
0 1 -2 12
0 0 1 2
Now add 2 times the 3rd row to the 2nd row (and replace the second row with this result)
1 -3 2 -21
0 1 0 16
0 0 1 2
Then add -2 times the 3rd row to the 1st row (and replace the first row with this result)
1 -3 0 -25
0 1 0 16
0 0 1 2
Finally, add 3 times the 2nd row to the 1st row (and replace the first row with this result)
1 0 0 23
0 1 0 16
0 0 1 2
The right hand column represents the solutions. So the solutions are x = 23, y = 16 and z = 2