2025 AMC 12B Problem 4

Setting $7a + b = 9b + a$ gives $6a = 8b,$ so $3a = 4b.$ Thus $a=4t$ and $b=3t.$ Because both are base-seven digits and $a\ne0,$ only $t=1$ works. Hence $a=4, b=3,$ and indeed $43_7 = 31 = 34_9.$ Therefore $a+b=7.$ Thus, the correct answer is A.