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Question Number 100792 by bramlex last updated on 28/Jun/20

Answered by bobhans last updated on 28/Jun/20

((x+xy)/y) = ((y+xy)/x) ⇒ x^2 +x^2 y = y^2 +xy^2   x^2 −y^2  = xy^2 −x^2 y ; (x−y)(x+y)=−xy(x−y)  ⇒ x+y = −xy . Then (1/x) + (1/y) = ((x+y)/(xy)) = −1 ★

$$\frac{{x}+{xy}}{{y}}\:=\:\frac{{y}+{xy}}{{x}}\:\Rightarrow\:{x}^{\mathrm{2}} +{x}^{\mathrm{2}} {y}\:=\:{y}^{\mathrm{2}} +{xy}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:=\:{xy}^{\mathrm{2}} −{x}^{\mathrm{2}} {y}\:;\:\left({x}−{y}\right)\left({x}+{y}\right)=−{xy}\left({x}−{y}\right) \\ $$$$\Rightarrow\:{x}+{y}\:=\:−{xy}\:.\:{T}\mathrm{hen}\:\frac{\mathrm{1}}{{x}}\:+\:\frac{\mathrm{1}}{{y}}\:=\:\frac{{x}+{y}}{{xy}}\:=\:−\mathrm{1}\:\bigstar \\ $$

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