If gcd(a,m)=1, then there is a unique inverse to a mod m. That is there is a unique solution to:
By the Euclidean Algorithm, you know there exist integers x and y such that
ax + my=1
Reduce mod m and we see
Uniqueness of this solution mod m comes from solving this diophantine equation. There are an infinite number of solutions to this equation ax +my = 1 if (a,m)=1 but they should all be congruent mod m. Alternatively if you know what a group is this is just the multiplicative group of units for the integers modulo m and as we all know inverses in a group are uniqe.
To solve your question, simply take the unique inverse that I found above and multiply it by b, multiplication is well defined, so this is the unique answer to the equation