Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.
Source: Turkey Team Selection Test 2017 P7
Tags: algebra, TST, functional equation
Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.