SOLUTION: Simplify completely: y over x - x over y divided by 2 over y -2 over x

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Simplify completely: y over x - x over y divided by 2 over y -2 over x      Log On


   

Question 135041: Simplify completely:
y over x - x over y divided by 2 over y -2 over x
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
y over x - x over y divided by 2 over y -2 over x
:
y%2Fx+-+x%2Fy
------
2%2Fy+-+2%2Fx
:
Place them over a common denominator
%28%28y%2Ay+-+x%2Ax%29%29%2F%28xy%29
---------
%28%282x+-+2y%29%29%2F%28xy%29
:
%28%28y%5E2+-+x%5E2%29%29%2F%28xy%29
---------
%28%282x+-+2y%29%29%2F%28xy%29
:
Invert the dividing fraction and multiply:
%28%28y%5E2+-+x%5E2%29%29%2F%28xy%29 * %28xy%29%2F%28%282x-2y%29%29
Cancel the xy's and you have:
%28%28y%5E2+-+x%5E2%29%29%2F%28%282x-2y%29%29
or
%28%28y%5E2+-+x%5E2%29%29%2F%282%28x-y%29%29
:
:
I failed to notice that y^2 - x^2 is the difference squares & can be factored to
%28%28y-x%29%28y%2Bx%29%29%2F%282%28x-y%29%29
Cancel (y-x) leaving us with:
%28x%2By%29%2F2
:
:
I can't believe that I keep screwing this up:
In order cancel y-x we have to change the sign in the denominator. Make it a -2
%28%28y-x%29%28y%2Bx%29%29%2F%28-2%28y-x%29%29
Cancel (y-x) leaving us with:
%28x%2By%29%2F%28-2%29 = -%28%28x%2By%29%29%2F2
:
Sorry if I got you thoroughly confused. Do you have any questions?