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Question Number 207249 by hardmath last updated on 10/May/24

$$\{\begin{array}{l}{\mathrm{x}}^2+{2\mathrm{y}}^2+\mathrm{xy}=37\\ {\mathrm{y}}^2+{2\mathrm{x}}^2+2\mathrm{xy}=26\end{array}\mathrm{find}:{\mathrm{x}}^2+{\mathrm{y}}^2=?$$

Answered by A5T last updated on 10/May/24

$$(x^2+y^2+xy)=21...\left(i\right)$$$$y^2-x^2-xy=11...\left(ii\right)$$$$\Rightarrow \left(i\right)+\left(ii\right):2y^2=32\Rightarrow y^2=16\Rightarrow y=\underset{-}{+}4$$$$y=4\Rightarrow x^2\underset{-}{+}4x-5=0\Rightarrow x=-5or1or5or1$$$$x^2=25or1\Rightarrow x^2+y^2=41or17.$$

Commented by hardmath last updated on 10/May/24

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}\mathrm{cool}$$

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