Prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$

Question

I'm meant to prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$.

I have a proof for this statement that uses the genus formula to show that such a curve must have genus $0$ and so is isomorphic to a $\mathbb{P}^1$, but I'm curious how one might prove this statement directly without reference to genus. Any help is welcome.