HMMT 2024 T6

Let $\mathbb{Q}$ be the set of rational numbers. Given a rational number $a \neq 0$, find, with proof, all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ satisfying the equation

$$f(f(x)+a y)=a f(y)+x$$

for all $x, y \in \mathbb{Q}$.