You can solve for x and y in terms of z, then let z take on the value of the parameter.
Subtracting the first equation from the second, you get
... x_2 +x_3 = 1
Subtracting that from the first equation gives you
... x_1 +2x_3 = 3
Let x_3 = t. Then your parametric equations are ...
[tex]x_1=3-2t\\x_2=1-t\\x_3=t[/tex]
