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Question Number 159469 by 0731619 last updated on 17/Nov/21

Commented by cortano last updated on 17/Nov/21

$$\sqrt{x}=uand\sqrt{y}=v$$$$\{\begin{array}{l}{u}^3+v^3=35\\ u^{2}v+{uv}^2=30\end{array}$$$$\{\begin{array}{l}{u}^3+v^3=35\Rightarrow {(u+v)}^3-3uv(u+v)=35\\ uv(u+v)=30\end{array}$$$$\Rightarrow {(u+v)}^3=125\Rightarrow u+v=5$$$$\Rightarrow uv=6\Rightarrow u(5-u)=6$$$$\Rightarrow u^2-5u+6=0\Rightarrow \{\begin{array}{l}u=3;v=2\\ u=2;v=3\end{array}$$$$\Rightarrow \{\begin{array}{l}x=9;y=4\\ x=4;y=9\end{array}$$

Answered by FongXD last updated on 17/Nov/21

$$\{\begin{array}{l}\sqrt{{\mathrm{x}}^3}+\sqrt{{\mathrm{y}}^3}=35\left(1\right)\\ \sqrt{{\mathrm{x}}^2\mathrm{y}}+\sqrt{{\mathrm{xy}}^2}=30\left(2\right)\end{array}$$$$\bullet \mathrm{take}\left(1\right)+3\left(2\right):$$$$\iff \sqrt{{\mathrm{x}}^3}+3\sqrt{{\mathrm{x}}^2\mathrm{y}}+3\sqrt{{\mathrm{xy}}^2}+\sqrt{{\mathrm{y}}^3}=125$$$$\iff {(\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})}^3=5^3,\Rightarrow \sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}=5\left(3\right)$$$$\mathrm{square}\mathrm{both}\mathrm{sides},\iff \mathrm{x}+\mathrm{y}+2\sqrt{\mathrm{xy}}=25$$$$\Rightarrow \mathrm{x}+\mathrm{y}-\sqrt{\mathrm{xy}}=25-3\sqrt{\mathrm{xy}}$$$$\left(1\right):\sqrt{{\mathrm{x}}^3}+\sqrt{{\mathrm{y}}^3}=35$$$$\iff (\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})(\mathrm{x}-\sqrt{\mathrm{xy}}+\mathrm{y})=35$$$$\iff 5(25-3\sqrt{\mathrm{xy}})=35$$$$\iff \sqrt{\mathrm{xy}}=6,\Rightarrow \mathrm{xy}=36$$$$\mathrm{if}\mathrm{you}\mathrm{want}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{x}\mathrm{and}\mathrm{y}\mathrm{which}$$$$\mathrm{satisfy}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{Eq}.\mathrm{above},\mathrm{just}\mathrm{substitute}\sqrt{\mathrm{x}}=\frac{6}{\sqrt{\mathrm{y}}}\mathrm{into}\left(3\right).$$

Commented by Rasheed.Sindhi last updated on 17/Nov/21

$$\mathcal{N}iceapproach!$$

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