Compute the ordered pair of positive integers $(x,y)$ such that \begin{align*} x^y+1&=y^x,\\ 2x^y&=y^x+7. \end{align*}
Let x^y=a,y^x=b. Then, a+1=b, 2a=b+7 so 2a=a+8 which means a=8,b=9 so x=2,y=3.
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