Question 138
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 $x, y \in \mathbb{Q}$.
$$
f(f(x)+a y)=a f(y)+x
$$
for all $x, y \in \mathbb{Q}$.