After a few eliminations (I won't show out my work because matrices are tedious to type in LaTeX), I got it to this point:
This seems to imply that in order for there to be any solutions, . (If , then our third equation is false.)
If , then the following equalities hold:
This implies that the infinite set of solutions is spanned by the line .
There seems to be no case in which there is a unique solution.