Question Number 208020 by Frix last updated on 02/Jun/24
$$\begin{cases}{\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\frac{\mathrm{14}}{\mathrm{625}}}\\{\sqrt{{x}}+\sqrt{{y}}=\mathrm{8}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{solutions}. \\ $$
Answered by efronzo1 last updated on 02/Jun/24
$$\:\:\Rightarrow\mathrm{x}+\mathrm{y}\:+\mathrm{2}\sqrt{\mathrm{xy}}\:=\:\mathrm{64}\: \\ $$$$\:\Rightarrow\:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{xy}}\:=\:\frac{\mathrm{14}}{\mathrm{625}}\:;\:\mathrm{x}+\mathrm{y}\:=\:\frac{\mathrm{14}}{\mathrm{625}}\mathrm{xy}\: \\ $$$$\:\Rightarrow\frac{\mathrm{14}}{\mathrm{625}}\left(\sqrt{\mathrm{xy}}\:\right)^{\mathrm{2}} +\:\mathrm{2}\sqrt{\mathrm{xy}}\:−\mathrm{64}\:=\:\mathrm{0} \\ $$$$\:\Rightarrow\:\left(\sqrt{\mathrm{xy}}−\mathrm{25}\right)\left(\mathrm{7}\sqrt{\mathrm{xy}}\:+\:\mathrm{800}\:\right)=\:\mathrm{0}\: \\ $$$$\:\Rightarrow\:\sqrt{\mathrm{xy}}\:=\:\mathrm{25}\:\mathrm{or}\:\:\underbrace{\nu} \\ $$