ABC26GN1989 · Surds and Indices

$2^{x}=8^{y+1} \Leftrightarrow 2^{x}=\left(2^{3}\right)^{y+1}=2^{(3 y+3)}$ $$\Leftrightarrow x=3 y+3 \Leftrightarrow x-3 y=3 \begin{aligned} 9^{y}=3^{x-9} & \Leftrightarrow\left(3^{2}\right)^{y}=3^{x-9} \\ & \Leftrightarrow 2 y=x-9 \Leftrightarrow x-2 y=9 \end{aligned}$$ Subtracting (i) from (ii), we get: $y=6$. Putting $y=6(i)$, we get $x=21$. $$\therefore \quad x+y=21+6=27 .$$