jmc

First, we find an expression for $h(x)$. From our definition of $h$, we have $h(4y-1)=2y+7$. So, if we let $x=4y-1$, so that $y=(x+1)/4$, we have $$h(x)=2\cdot \frac{x+1}{4}+7=\frac{x+1}{2}+7.$$ Setting this equal to $x$ gives $$x=\frac{x+1}{2}+7.$$ Multiplying both sides by 2 gives $2x=x+1+14$, so $x=\boxed{15}$.