Basic algebra problem: 1 x </mfrac> + 1

Arraryeldergox2 2022-06-26 Answered
Basic algebra problem: 1 x + 1 y 1 x 2 1 y 2
x , y R , x 2 y 2 , x y 0
Now I know the result is: x y y x , but I am not sure how to get it, I get into a mess like this: = x + x 2 y y 2 x y = x ( x y ) + x 3 y 3 y ( x y ) x y = ? which doesn't seem to help me much. Halp please.
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Answers (2)

Jake Mcpherson
Answered 2022-06-27 Author has 23 answers
1 x + 1 y 1 x 2 1 y 2 = 1 x + 1 y ( 1 x + 1 y ) ( 1 x 1 y ) = 1 1 x 1 y = 1 y x x y = x y y x

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aligass2004yi
Answered 2022-06-28 Author has 7 answers
First write 1 x + 1 y = y x y + x x y = x + y x y
Then write 1 x 2 1 y 2 = y 2 x 2 y 2 x 2 x 2 y 2 = y 2 x 2 x 2 y 2 = ( y x ) ( x + y ) x 2 y 2
Therefore
1 x + 1 y 1 x 2 1 y 2 = x + y x y ( y x ) ( x + y ) x 2 y 2 = x + y x y x 2 y 2 ( y x ) ( x + y ) = ( x + y ) x 2 y 2 x y ( x + y ) ( y x ) = x y y x

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