algebra precalculus - Why is that when we are adding to both sides of an equation we don't add to every single term?

We have all heard the age old expression of what you do to one equation, you must do this to the other equation.
Lets take an example equation of $$3x - 2 = 4x +5$$
when solving this equation you would add two to both sides and get

$$3x = 4x + 7$$
my question is why is that when we are adding to both sides that we don't add to every single term, but when we do a multiplication or division operation we would multiply / divide every term.
Also another question, in essence when we are doing something to both sides can we treat each side of the equation like a term. So in essence for the equation when we are dividing by both sides is it basically: $$1/2(3x-2) = 1/2(4x+5)$$ , meaning when we do something to both sides do we treat the equation like a term? What exactly is happening behind the scenes? Why is it that you multiply every term but you don't add every single term?