This is meant as a note. I will definitely expand this article later.
I hope this method has not been discovered before :(.
Say is a recursion relation. Then
is the intersection point of
and
.
For example, if we have , then
is the y-coordinate of the intersection of
and
.
Nice post, but it will be more illuminating if you can give an example where solving the simultaneous equations $x=f(y)$ and $y=f(x)$ is easier than solving $x=f(x)$