Show that a homomorphism from $SL_2(\mathbb{Z})$ to $ SL_2(\mathbb{F}_p)$ is surjective

Question

The homomorphism sends the entries of matrices in $SL_2(\mathbb{Z})$ to their congruence classes mod $p$. After a lot of work I could prove that a matrix $$\begin{pmatrix}a & b\\\ c & d\end{pmatrix}$$ in $ SL_2(\mathbb{F}_p)$ has a preimage in $SL_2(\mathbb{Z})$ if $a$ and $c$ are relatively prime and I can compute it. But I can't prove the general case and the method I used to prove the preceding assertion was so complicated that it can't be the right way to approach the problem. So there has to be another, more elegant, approach. I attacked the problem head on and ended up with a system of diophantine equations which was difficult to solve as one of the equations was quadratic.