SOLUTION: If 1/x - 1/y = 1/(x+y) Find (y/x + x/y)

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Question 1206591: If 1/x - 1/y = 1/(x+y)
Find (y/x + x/y)
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52884)   (Show Source): You can put this solution on YOUR website!
.
If 1/x - 1/y = 1/(x+y), find (y/x + x/y).
~~~~~~~~~~~~~~~~~~~~~~


        I think that the correct formulation is  DIFFERENT.

        It should be 
            If 1/x - 1/y = 1/(x+y), find (y/x - x/y).

        I will solve it below in this formulation. 


From 

    -  =     (1)

you get

    = ,

   (y-x)*(y+x) = xy,

    y^2 - x^2 = xy.


Divide both sides by xy

     -  = 1.


At this point, the solution to the problem is complete.


ANSWER.  If   -  = ,  then   -  = 1.

Solved.



Answer by Edwin McCravy(20064)   (Show Source): You can put this solution on YOUR website!
Ikleyn changed the problem. Here is the original problem:

If , find  

To be defined we must require 

,







Solve for y by the quadratic formula:











Dividing both sides by x



Using +

, which is positive, which

means this is the case when x and y have the same sign.

Taking reciprocals of both sides:



Therefore



Using -

, which is negative, which

means this is the case when x and y have opposite signs.

Every sign before  in the above will change, so the last step will
be:



So if

 then 

If x and y have the same sign, the sign of  will be + 
and if x and y have opposite signs, the sign of  will be -.

Edwin
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