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

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

Answered by efronzo1 last updated on 02/Jun/24

$$\Rightarrow \mathrm{x}+\mathrm{y}+2\sqrt{\mathrm{xy}}=64$$$$\Rightarrow \frac{\mathrm{x}+\mathrm{y}}{\mathrm{xy}}=\frac{14}{625};\mathrm{x}+\mathrm{y}=\frac{14}{625}\mathrm{xy}$$$$\Rightarrow \frac{14}{625}{\left(\sqrt{\mathrm{xy}}\right)}^2+2\sqrt{\mathrm{xy}}-64=0$$$$\Rightarrow (\sqrt{\mathrm{xy}}-25)(7\sqrt{\mathrm{xy}}+800)=0$$$$\Rightarrow \sqrt{\mathrm{xy}}=25\mathrm{or}\underbrace{\nu }$$

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