what is the result when you add 0 to both sides of an equation?

I asked this already, but I would like a deeper explination

2 answers

0 is the additive identity. So, by definition.
x+0 = 0

Then there is what is commonly called the additive property of equality:
equals added to equals are equal.
That is, if you have
x = x
then x+y = x+y
So, if y=0, x+0 = x+0 ... x=x
When 0 is added, nothing changes.