Convention - Dual $\pm$ Signs

Question

Say I have two equations like $$z+w=x+y$$ $$z-w=x-y$$ Is there any way I can combine these into a statement like $$z\pm w=x\pm y$$ I know the above statement is not correct as that entails 4 different equations when I only am asserting the above two, however is there any way like that for me to combine the two very similar statements.
I am using this in a paper I am writing but the two equations I have take up and annoying amount of room for the fact that they only differ by plus/minus sign.

Answer 1

Some authors, in some contexts, follow the convention that an equation with any number of $\pm$ signs in it stands for exactly two equations: one where they all become $+$, and one where they all become $-$.

One can also use a $\mp$ sign in combination with this to stand for a sign that goes the opposite of the other one(s).

With this convention, "$z\pm w=x\pm y$" will mean exactly what you want to express.

Answer 2

If you rewrite your equation, you get :

$z+w = x+y \Leftrightarrow z-x = y-w$

$z-w = x-y \Leftrightarrow z-x = -(y-w)$

Hence $z-x = \pm (y-w)$