# Proof that expression divided by opposite equals $-1$

I'm reviewing some basic algebra and I came across a statement saying that: $$\frac{a - b} {b - a} = -1$$ Plugging in a few values this appears to be true.

However, I have been unable to find a formal proof for this. Neither do I know what is this called? Could someone provide additional information on this matter?

• Factor out a $-1$ from either the numerator or the denominator. – poweierstrass Jan 8 '17 at 15:17
• We have $$P= \frac {a-b}{b-a} =\frac {a-b}{-(a-b)} =-1$$ – Rohan Jan 8 '17 at 15:18

## 4 Answers

We have $$P= \frac {a-b}{b-a} =\frac {a-b}{-(a-b)} =-1$$ We can also write it as $$P=\frac {a-b}{b-a} =\frac {-(b-a)}{b-a} =-1$$ Hope it helps.

It was quite easy you have just to take $(-)$ minus common from numerator or denominator $$\frac{a-b}{b-a}=\frac{a-b}{-(a-b)}=\frac1{-1}=-1$$ $$**OR**$$ $$\frac {a-b}{b-a} =\frac {-(b-a)}{b-a} =\frac{-1}1=-1.$$

since we get by the minus sign $$-(a-b)=b-a$$

$$\frac{(a-b)}{(b-a)}=\frac{(a-b)}{-(a-b)}=\frac{1}{-1}=-1$$