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# English & Math Ref
1 Substitute x = b - a into a + x gives us substitution, addition is commutative, definition of negative element, addition is associative, definition of zero element
a + b - a = b + a - a = b + 0 = b
2 Therefore from (1), x = b - a is a solution to the question's equation
3 Suppose there is an arbitrary solution x1 by assumption
a + x1 = b
4 Adding the negative of a to both sides you can add the same term to both sides of an equation as the equality of lhs = rhs means that the resultant sum of the lhs and the term added and rhs and the term added are equal to each other
a + x1 - a = b - a
5 Re-arranging: addition is commutative, definition of negative element, definition of zero element
x1 + a - a = b - a = x1 + 0 = b - a x1 = b - a
6 Therefore if a solution exists, it will be = b - a and accordingly is unique as there is only one solution
7 Alternatively we can do steps (3..5) for a second arbitrary x2 and then show proving uniqueness by showing two arbitrary items are equal to each other
x1 = x2
8 However, we proved in (2) that b - a is a solution and therefore it is the unique solution to the equation.