You know, I'm looking at this again and I'm noting that we may use equations 1 and 3 to come up with a new equation in just b and d, which we may use together with the second equation to get values for b and d unambiguously. Then note that if we are to be able to get a consistent solution equations 1 and 3 must be the same (as a and c both have the same coefficients.) So we can only find one of them.
Thus there is one free parameter, either a or c.
PS Sorry about all the edits!