Could Someone Explain the Intermediary Steps?

[ReviewStudent]

Ok, so this review packet I am working from does not fully explain their answers, and I am having difficulty following some of them. Any help would be greatly appreciated.

Here is the expression:

(x^-1 – y^-1) / (x-y)^-1

Now, I understand the next step:

(1/x - 1/y) / (1/x-y)


But how do they get to this:

(x-y)(y-x)/xy

I tried working backwards, but I guess mu understanding of fractions is a bit rusty. Am I supposed to multiply by (1/x-y)? What is the best way to approach a problem like this? Thanks in advance.

[stapel_eliz]

(x^-1 – y^-1) / (x-y)^-1

Now, I understand the next step:

(1/x - 1/y) / (1/x-y)

But how do they get to this:

(x-y)(y-x)/xy

What did you get when you converted the subtracted fractions on top to a common denominator and combined them? What did you get when you flipped the bottom fraction and multiplied it against the top fraction?

[ReviewStudent]

Ok, I now understand their simplification.

(x^-1 – y^-1) / (x-y)^-1
= (1/x - 1/y) / (1/x-y)
= (y-x/xy) / (1/x-y)
= (y-x/xy)(x-y/1)
...

But then wouldn't you continue, and go:

= 2xy - y^2 - x^2 / xy
= -y^2 - x^2 +2

?

My book stops at the ellipse. Why? What makes that simplification better?

Oh, and thanks for the clarification as to what I should do. I was unnecessarily confused by the fractions xD.

[stapel_eliz]

... = (y-x/xy)(x-y/1)

But then wouldn't you continue, and go:

= 2xy - y^2 - x^2 / xy

How did you go from here (missing step above displayed below):

. . . . .

. . . . .

. . . . .

...to an expression with no denominator? Nothing cancels (or you would have cancelled it out before multiplying out), so where did the denominator go? And how did the numerator change in the way that it did?

Please show those missing steps. Thank you!