2^x+2^y=8=2^3; 3^x+3^y=2*3^2.
The second equation seems to hold the clue because there are two terms on the left and if they are equal we have 2*3^x=2*3^2, and x=2 so y=2. Now we turn to the first equation: 2*2^x=2*2^2, so x=y=2; therefore the solution is x=y=2.