First observe that you in fact only have two equations as the third is a multiple of the first.

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.

RonL