jmc

We substitute $a=x^y$ and $b=y^x$ to form the equations

$$\begin{array}{rl}a+1& =b,\\ 2a& =b+7.\end{array}$$

Subtracting the first equation from the second, we obtain $a-1=7$, so $a=8$.

Substituting this into the first equation, we find $b=9$.

We see from $x^y=8$ and $y^x=9$ that the solution is $(x,y)=\boxed{(2,3)}$.