Pascal's triangle induction proof

Question

I am trying to prove $$\binom{n}{k} = \binom{n}{k-1}\frac{n-k+1}{k}$$ for each $k \in \{1,...,n\}$ by induction. My professor gave us a hint for the inductive step to use the following four equations:
\begin{align*} \binom{n + 1}{k} & = \binom{n}{k} + \binom{n}{k - 1}\\ \binom{n + 1}{k - 1} & = \binom{n}{k - 1} + \binom{n}{k - 2}\\ \binom{n}{k} & = \binom{n}{k - 1}\frac{n - k + 1}{k}\\ \binom{n}{k - 1} & = \binom{n}{k - 2}\frac{n - k + 2}{k - 1} \end{align*} I keep getting stuck in the inductive step. I was hoping someone could help me.