I think you are looking to find b such that a + b = 1 mod n. The problem is to find b such that a x b = 1 mod n.
That only has a solution if a is relatively prime to n, and generally the best way to find that solution is to use the extended Euclidean algorithm [1].
Briefly because a and n are relatively prime, there exists integers x and y such that ax + ny = 1 [2]. The extended Euclidean algorithm finds x and y. But ax + ny mod n is just ax mod n, and so we have ax = 1 mod n. The b we seek is x.
[1] https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm