Suppose a,b e G where G is a group under addition.
Then, the equation x + a = b has a unique solution in G. The solution is
x = b - a.
Suppose the equation x + a = b has two solutions. Let the solutions be
x = b - a and y = b - a.
Now, by transitivity of equality, x = y. THus there is a unique solution in G.
How does this look?