Of course, anyone who has done undergraduate linear algebra knows to look to the determinant in case some shit goes down. For the general problem space, a zero determinant means that one equation is just a multiple of the other. A solution could still exist in this case. Consider:

a => x+1, y+1
b => x+2, y+2
x = 4, y = 4 (answer: 2 tokens)

The following has no solution:

a => x+2, y+2
b => x+4, y+4
x = 3, y = 3

I thought of all this, and instead of coding the solution, I just checked if any such cases were in my input. There weren't, and I was home free.

I'm looking at my code again and I'm pretty sure that was all unnecessary.