Suppose our ring $R = k[t]$. Let $M$ be the $R$-module $R[x]/(tx - t)$. Prove that $M$ isn't flat.

Question

I proved this problem before by the following theorem. Over PID Flat iff torsion free. Then, from this fact, we know that M isn't flat as it has torsion element. In the problem we were given hint that to consider the following sequence $0 \rightarrow (t) \rightarrow R$. I am guessing, we then proceed to show that $M \otimes_R (t)$ doesn't inject into $M \otimes_R R$. I couldn't show that this doesn't inject. If someone could help that would be nice.