Prove $(-x)y=-(xy)$ using axioms of real numbers

Question

Working on proof writing, and I need to prove

$$(-x)y=-(xy)$$

using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am unsure how to prove it.

Answer 1

You could prove that $(-x)y$ and $-(xy)$ are both the additive inverse of $xy$. Then use its uniqueness ($5^{th}$ axiom).

$$xy + (-x)y = (x+(-x))y = 0y = 0$$

Notice that $0y = 0$, because : $$0y = (0+0)y = 0y + 0y$$ and if you add the additive inverse of $0y$ to both of the sides you come up with : $$(-0y) + 0y = (-0y) + 0y + 0y$$ i.e. $$0 = 0y$$

Answer 2

First start by proving : -x = (-1)x. Proof: x+(-1)x = (1+(-1))x = 0x = 0. The proposition follows from uniqnes off negative off a number. Now you can use this to prove your statement. Proof: (−x)y = ((-1)x)y = (-1)(xy) = -xy