Answer
$ad+b=bc+d$
Work Step by Step
We are given the functions:
$$\begin{align*}
f(x)&=ax+b\\
g(x)&=cx+d.
\end{align*}$$
In order to have $f\circ g=g\circ f$ the following equations must be true for all $x$:
$$\begin{align*}
(f\circ g)(x)&=(g\circ f)(x)\\
f(g(x))&=g(f(x))\\
f(cx+d)&=g(ax+b)\\
a(cx+d)+b&=c(ax+b)+d\\
acx+ad+b&=acx+bc+d\\
ad+b&=bc+d.
\end{align*}$$
Therefore the necessary and sufficient condition on the constants $a$, $b$, $c$ and $d$ is
$$ad+b=bc+d.$$
