How to prove these ?

Question

If $a,b ϵ R$ then prove that if $a+b=0$ then $b=-a$ ?

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Answer 1 · 2017-09-08T00:15:09

Add $-a$ to both members of the equality:

$-a+ (a+b) = -a$

using the commutative property of the sum of two real numbers:

$-a + (a+b) = (-a+a) + b$

by the definition of opposite of a real number:

$(-a+a) + b= 0 + b$

and as zero is the identity element of the sum:

$0+b = b$

Then by the transitive property of the equality of two real numbers:

$b = -a$