Prove that an equation with a zero divisor can't have a unique solution

Let R be a commutative ring, and suppose b is a zero divisor in R. Let c be any element in R. Prove that the equation bx = c can never have a unique solution in R. (If it has any solution, there is more than one solution.)

I know that if b is a zero divisor, there must be some c != 0 in R where bc = 0, and that you can't cancel a zero divisor from a product. I guess that if there is a solution to bx = c, there must also be a y such that by = c, where x != y. I could set the two equal to each other: bx = by, and then simplify to b(x-y) = 0. I'm not sure where to go from here, or even if this was the right way to start the proof.