jmc

The slope of the line through $(7,8)$ and $(9,0)$ is $\frac{8-0}{7-9}=\frac{8}{-2}=-4$. Thus, the line has equation $y=-4x+b$ for some $b$. Since $B(9,0)$ lies on this line, we have $0=-4(9)+b\Rightarrow b=36$, and thus the equation of the line is $y=-4x+36$.

To determine the point of intersection between the lines having equations $y=-4x+36$ and $y=2x-10$, we set the two expressions for $y$ equal to each other and solve for $x$. We have $-4x+36=2x-10\Rightarrow x=\frac{23}{3}$. It follows that $y=2x-10=2\left(\frac{23}{3}\right)-10=\frac{46}{3}-\frac{30}{3}=\frac{16}{3}$.

Thus, $P=(\frac{23}{3},\frac{16}{3})$ and $a+b=\frac{23}{3}+\frac{16}{3}=\frac{39}{3}=\boxed{13}$.