jmc

Converting the two expressions to base $10$, it follows that the given positive integer is equal to $9A+B$ and is also equal to $7B+A$. Setting these two expressions equal, we have that $$9A+B=7B+A⟹8A=6B⟹4A=3B.$$ Thus, $B$ is divisible by $4$. Since $B$ is a digit in base $7$, it follows that $B$ is either equal to $0$ or $4$. However, we can discard the case $B=0$, since its base $7$ representation is no longer a two-digit number. Thus, $B=4$ and $A=3$. In base $10$, the number is $9\cdot 3+4=7\cdot 4+3=\boxed{31}.$