The functions $f$ and $g$ are defined as $f(x)=\frac{5}{2}x+1$ and $g(x)=-\frac{1}{4}x+5$. If the function $h$ is defined as $h(x)=f(x)+g(x)$, what is the $x$-coordinate of the $x$-intercept of the graph of $y=h(x)$ in the $xy$-plane?