Next step?

[jtw2e2]

I have tried to simplify the following expression and am stuck:

[(x/y)-(y/x)] / [(1/x^2)-(1/y^2)]

I got this far:

[ (x^2 - y^2) / (xy) ] times [ ((x^2 )(y^2)) / (y^2 - x^2)

Is that correct so far? Would someone please direct me through the next step? Thank you.

 

[mmm4444bot]

… Is that correct so far? …

It looks good, to me.

Next, factor x^2 - y^2 and factor y^2 - x^2.

Also, do you see any cancellations in the ratio (x^2 y^2)/(xy) ?

 

[mmm4444bot]

Here's a fact that you might not realize.

y - x = -(x - y)

You can use this fact, after you factor the two differences of squares.

 

[jtw2e2]

Re:

Here's a fact that you might not realize.

y - x = -(x - y)

You can use this fact, after you factor the two differences of squares.

I don't realize much :?

But here's what I got:

xy / -1

I don't know if it's right. When I turned (y-x) into -(x-y), do I have to multiply both the numerator and denominator by negative 1?

 

[jtw2e2]

Re: Re:

do I have to multiply both the numerator and denominator by negative 1?

Nevermind, I think I understand why it is ok to multiply by the -1 because the order is being switched.

xy / -1 = -xy

I think that's it!

 

[mmm4444bot]

… -xy

I think that's it!

Excellent! You realize much more than you know, Grasshopper.

(Seriously, you're operating at a level substantially higher than most people seeking help here.)

Another fact that's often helpful:

\(\displaystyle -\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}\)

This is actually an abbreviation for the following.

\(\displaystyle (-1) \cdot \frac{a}{b} = \frac{(-1)}{1} \cdot \frac{a}{b} = \frac{1}{(-1)} \cdot \frac{a}{b}\)

In other words, that negation symbol is shorthand for a factor of -1, and it can float around, as we please, out in front of a ratio, or in the numerator, or in the denominator.

Cheers ~ Mark

 

[jtw2e2]

Re:

....

Cheers ~ Mark

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Thanks for your help Mark. I can go to sleep in peace now that I finished that one. :lol: