Hi! Thank you guys for responding. I solved it! I did it like this:

The base 7 number can be "abc_7", and a, b, and c are all digits from 1 to 6, except for b which can also be 0. Then you can expand it to be 49a + 7b + c. Then when it is reversed, you get cba. expanding in base 9, cba_9= 81c+ 9 b + a. The two expressionss are equal, so you can use the equation 49a + 7b + c = 81c + 9b + a. Then you move everything to one side and get 80c + 2b - 48a = 0, and you can simplify and get 40 c+ b- 24a = 0. Then we can solve for b and get b = 24a - 40c = 8(3a - 5c). This means that b is divisible by 8, but it is also a base 7 digit so it has to be 0. then you have 3 a = 5c sp a is divisible by 5 and c is divisible by 3. They cant be 0 because they are both left digits, so they are 5 and 3. This means that the answer is 503_7.

Hooray! Thank you again for helping me :)