So we want to find the general solution of the system with augmented matrix:
1 2 1 -1 3
0 0 1 -1 2
Put y=a, w=b, then z=2+b, and x=1-2a. Which you can get by by solving
the third equation, which in full form is:
z - w = 2
and then the first equation which is now:
x + 2y + z - w = 3.