Let $h(4x-1)=2x+7$. For what value of $x$ is $h(x)=x$? Let $h(4x-1)=2x+7$. For what value of $x$ is $h(x)=x$?
If $h(a)=a$, then $4x-1=2x+7$ which implies $x=4$. So $a=15$ when I substitute $x=4$ into both linear equations. Is the value of $x$ $15$?

The posted proof looks good. For an alternative one, let $4x-1=y \iff x = \frac{y+1}{4}\,$, then:
$$\,h(y) = h(4x-1)=2x+7=2\,\frac{y+1}{4}+7=\frac{y}{2}+\frac{15}{2}\,$$
Therefore $\,h(x)=\frac{x}{2}+\frac{15}{2}\,$, and $h(x)=x \iff x = \frac{x}{2}+\frac{15}{2} \iff x=15\,$.
