^good point, I wasn't looking for it to be done in its entirety, however, as I've discovered this problem is a lot simpler than I thought it was.. i picked at it and eventually got the proper answer, although im not entirely aware of the theory behind the work itself
i took 2x + 3y + 7z = -5 and let y = s and z = t
with that, you can figure out that:
x= -3/2s - 7/2t - 5/2
and so in vector parametric form (horizontally):
(x,y,z) = (-5/2, 0, 0) + s(-3/2, 1, 0) + t(-7/2, 0, 1)