The book is trying to show me that reducing a fraction or adding two fractions sometimes only requires that -1 be factored from one or more denominators. Here is the example they give:

$\displaystyle \frac{y-x}{x-y}=\frac{y-x}{-(-x+y)}=\frac{y-x}{-(y-x)}=\frac{1}{-1}=-1$

1.In the first step here I understand factoring out the "-" (the -1 value) because if the variable does not have a number coefficient we can assume its one so this part is fine.

$\displaystyle \frac{y-x}{-(-x+y)}$

2.Here$\displaystyle \frac{y-x}{-(-x+y)}$

once we distribute the "-" we get a negative y and a positive x, therefore we can rearrange the denominator to match the numerator in $\displaystyle \frac{y-x}{-(y-x)}$

3.Next we have $\displaystyle \frac{1}{-1}$ Now here is where I have a problem how can we know the values of both of these are 1 and -1? I am not certain how this is accomplished.

Does my logic up to 3's question look correct also?