Explanation for $k{n \choose k - 1} = (k-1){n \choose k - 1} + {n \choose k - 1}$

Question

Does anyone have a good explanation for why this makes sense? I am trying to remember this fact in a proof but simply can't seem to make a good explanation for it.

NOTE: I have already shown that $k{n \choose k - 1} = (k-1){n \choose k - 1} + {n \choose k - 1}$ through mathematically reducing the RHS to the LHS. But I would like a meaning/explanation in words for why it makes since that the LHS would still equal the RHS after manipulation. My professor mentioned something about if we take away a k, then we have to add it back or something.

Any attempt to explain is great! Thanks!

Answer 1

$k=(k-1)+1$, now multiply both sides by $\binom{n}{k-1}$