Finding parametric representations fo the parts of the plane

Question

How would I find parametric representations for the plane:
$2x+3y+z=4$ for $0 \leq x + y + z \leq 7$ and $2 \leq x-y \leq 4$?
I can do simple ones where only $x,y$ are restricted independently (forms a rectangle in the $x-y$ plane, but how would I go about doing this one (when all three variables are involved)?

Answer 1

Let $x-y=u$ and $x+y+z=t$. Then you can find $x$, $y$ and $z$ in terms of $u$ and $t$ by Crammer's Rule.

$$(x,y,z)=\left(\frac{2u-t+4}{3},\frac{-u-t+4}{3},\frac{-u+5t-8}{3}\right),$$

where $2\le u\le 4$ and $0\le v\le7$.