  Assuming a modular version of Schanuel's conjecture and the modular
Zilber-Pink conjecture, we show that the existence of generic solutions of
certain families of equations involving the modular $j$ function can be reduced
to the problem of finding a Zariski dense set of solutions. By imposing some
conditions on the field of definition of the variety, we are also able to
obtain unconditional versions of this result.
