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Question Number 208020 by Frix last updated on 02/Jun/24

{ (((1/x)+(1/y)=((14)/(625)))),(((√x)+(√y)=8)) :} Find all solutions.

$$\begin{cases}{\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\frac{\mathrm{14}}{\mathrm{625}}}\\{\sqrt{{x}}+\sqrt{{y}}=\mathrm{8}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{solutions}. \\ $$

Answered by efronzo1 last updated on 02/Jun/24

⇒x+y +2(√(xy)) = 64 ⇒ ((x+y)/(xy)) = ((14)/(625)) ; x+y = ((14)/(625))xy ⇒((14)/(625))((√(xy)) )^2 + 2(√(xy)) −64 = 0 ⇒ ((√(xy))−25)(7(√(xy)) + 800 )= 0 ⇒ (√(xy)) = 25 or { ((x+y = 14 )),((xy = 625)) :}⇒ x(14−x) = 625 x^2 − 14x +625 = 0 x = ⇒ { (( ( / ))),(( ( / ))) :}

$$\:\:\Rightarrow\mathrm{x}+\mathrm{y}\:+\mathrm{2}\sqrt{\mathrm{xy}}\:=\:\mathrm{64}\: \\ $$$$\:\Rightarrow\:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{xy}}\:=\:\frac{\mathrm{14}}{\mathrm{625}}\:;\:\mathrm{x}+\mathrm{y}\:=\:\frac{\mathrm{14}}{\mathrm{625}}\mathrm{xy}\: \\ $$$$\:\Rightarrow\frac{\mathrm{14}}{\mathrm{625}}\left(\sqrt{\mathrm{xy}}\:\right)^{\mathrm{2}} +\:\mathrm{2}\sqrt{\mathrm{xy}}\:−\mathrm{64}\:=\:\mathrm{0} \\ $$$$\:\Rightarrow\:\left(\sqrt{\mathrm{xy}}−\mathrm{25}\right)\left(\mathrm{7}\sqrt{\mathrm{xy}}\:+\:\mathrm{800}\:\right)=\:\mathrm{0}\: \\ $$$$\:\Rightarrow\:\sqrt{\mathrm{xy}}\:=\:\mathrm{25}\:\mathrm{or}\: \\ $$$$ \begin{cases}{\mathrm{x}+\mathrm{y}\:=\:\mathrm{14}\:}\\{\mathrm{xy}\:=\:\mathrm{625}}\end{cases}\Rightarrow\:\mathrm{x}\left(\mathrm{14}−\mathrm{x}\right)\:=\:\mathrm{625}\: \\ $$$$\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{14x}\:+\mathrm{625}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{x}\:=\: \\ $$$$ \Rightarrow\:\begin{cases}{ \frac{ }{ }}\\{ \frac{ }{ }}\end{cases} \\ $$$$ \\ $$

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