Is $xRy \iff x+y = 0$ an equivalence relation? $R$ is a relation on real numbers. $xRy \iff x+y = 0 $.
Is it an equivalence relation?
My answer is no
proof:
-(Reflexive) let $x = a$ , $aRa \iff 2a=0$.
 Since $2a = 0$ doesn't hold for every real number $a$, $R$ is not reflexive.
Since $R$ isn't reflexive, $R$ is not an equivalence relation.
Is my reasoning correct ?

It is not only correct, it is almost the most straightforward approach you could take. Nicely done!
The only improvement I would suggest is that you demonstrate a particular real number $a$ for which $a\:R\:a$ fails to hold. For example, you could just say: "Since $1+1=2\ne 0,$ then $1\:R\:1$ fails to hold, and so $R$ is not an equivalence relation on the real numbers."
