There is no unique solution in general. That is, infinitely many possible solutions.
In this case, you have TWO variables, and ONE equation. So you could pick any value for X, substitute it into the equation, and then the value for Y is known, as long as b is not equal to zero. That is, you would compute
Y = (c - a*X)/b
Likewise, you could pick any value for Y, and then solve for X, as a function of Y. Again, that requires only that a is non-zero.
X = (c - b*Y)/a
Either approach is equally valid. There are also ways you can write a solution in the form of one that minimizes the norm of the vector [X,Y]. That is just another way to choose one of the infinitely many solutions.
So does a solution exist? Well, no. And, yes. But just not "a" solution. Any solution you desire.