How can theorem 9 algebra be proved Hi how can theorem 9 be

How can theorem 9 (algebra) be proved?

Hi! how can theorem 9 be proved? theorem 9 states that for any numbers a and b, there is a unique solution to the equation a+x=b. definition: b-a=b+(-a)

Solution

a+x=b

The above is an equation. In an equation the equals can be added or subtracted. Then the results are also equal.

So, subtract -a from both sides :

a+x-a = b-a.

x+a-a=b-a,    as a+x=x-a, addition is commutative.

x= b-a . And this is the solution for x.

If the solution x =b-a is not unique, then there must be another solution x\' other than b-a.Then substitute x\' in the original equation , and you get:

a+x\'=b.Or x\' = b-a. But b-a = x also. Therefore x=x\'. So, the solution is unique.

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