Can we deduce the exactness of the pervious modules sequence if the localized exact modules sequence is exact?

Question

My question comes from a proposition: if M is flat, will $S^{-1}M$ be flat? Where $S = R - \mathfrak{p}$, $\mathfrak{p}$ is a prime ideal. Since localization keeps the exactness, I find that what I have to do is to prove if $0 \to S^{-1}M \to S^{-1}N \to S^{-1}L \to 0$ is exact then $0 \to M \to N \to L \to 0$ is flat. The first step to prove this is that I have to define the morphisms in the second sequence. However, I don't know how to.