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Question Number 216842 by ArshadS last updated on 22/Feb/25

$$\mathrm{Find}\mathrm{all}\mathrm{pairs}\mathrm{of}\mathrm{positive}\mathrm{integers}\mathrm{x},\mathrm{y}\mathrm{that}\mathrm{satisfy}$$$$\mathrm{the}\mathrm{system}$$$$\mathrm{xy}+\mathrm{x}+\mathrm{y}=71$$$${\mathrm{x}}^2\mathrm{y}+{\mathrm{xy}}^2=880$$

Answered by Frix last updated on 22/Feb/25

$$xy+(x+y)=71$$$$xy(x+y)=880$$$$xy=71-(x+y)$$$$(71-(x+y))(x+y)=880$$$$71(x+y)-{(x+y)}^2=880$$$${(x+y)}^2-71(x+y)+880=0$$$$(x+y)=\frac{71}{2}\pm \frac{39}{2}$$$$(x+y)=55\vee (x+y)=16$$$$\Rightarrow (x+y)=55\wedge xy=16\mathrm{impossible}$$$$\vee (x+y)=16\wedge xy=55$$$$\Rightarrow x=5\wedge y=11\vee x=11\wedge y=5$$

Commented by ArshadS last updated on 22/Feb/25

$$\mathbb{N}\mathrm{ice}\mathrm{sir}!$$

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