Solve-for-natural-numbers-x-2-y-2-x-y-3xy-

Question Number 149266 by mathdanisur last updated on 04/Aug/21

$${Solve}\:{for}\:{natural}\:{numbers}: \\ $$$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+{x}\:+\:{y}\:=\:\mathrm{3}{xy} \\ $$

Answered by nimnim last updated on 04/Aug/21

⇒(x+y)^2 +(x+y)=5xy ⇒(x+y)(x+y+1)=5xy for x,y∈N case.1 x+y=xy and x+y+1=5 ⇒x+y=4 ⇒(x,y)=(2,2) case.2 x+y=5 and x+y+1=xy ⇒ (x,y)=(2,3) or (3,2) ∴solution set={(2,2),(2,3),(3,2)}

$$\Rightarrow\left({x}+{y}\right)^{\mathrm{2}} +\left({x}+{y}\right)=\mathrm{5}{xy} \\ $$$$\Rightarrow\left({x}+{y}\right)\left({x}+{y}+\mathrm{1}\right)=\mathrm{5}{xy} \\ $$$${for}\:{x},{y}\in{N} \\ $$$${case}.\mathrm{1}\:\:\:\:{x}+{y}={xy}\:\:\:\:{and}\:\:{x}+{y}+\mathrm{1}=\mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow{x}+{y}=\mathrm{4} \\ $$$$\Rightarrow\left({x},{y}\right)=\left(\mathrm{2},\mathrm{2}\right) \\ $$$${case}.\mathrm{2}\:\:\:\:{x}+{y}=\mathrm{5}\:\:{and}\:{x}+{y}+\mathrm{1}={xy} \\ $$$$\Rightarrow\:\left({x},{y}\right)=\left(\mathrm{2},\mathrm{3}\right)\:{or}\:\left(\mathrm{3},\mathrm{2}\right) \\ $$$$\therefore{solution}\:{set}=\left\{\left(\mathrm{2},\mathrm{2}\right),\left(\mathrm{2},\mathrm{3}\right),\left(\mathrm{3},\mathrm{2}\right)\right\} \\ $$$$\:\: \\ $$

Commented by mathdanisur last updated on 04/Aug/21