solve: 2x+3y=2 x+y=5xy


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Question Number 171837 by Mikenice last updated on 21/Jun/22

$s o l v e :$ $2 x + 3 y = 2$ $x + y = 5 x y$
	Answered by Rasheed.Sindhi last updated on 21/Jun/22
	$2x+3y=2⇒y=((2−2x)/3) x+y=5xy⇒x+((2−2x)/3)=5x(((2−2x)/3)) ⇒3x+2−2x=10x−10x^2 ⇒10x^2 −9x+2=0 10x^2 −5x−4x+2=0 5x(2x−1)−2(2x−1)=0 (2x−1)(5x−2)=0 x=(1/2) ∣ x=(2/5) •y=((2−2x)/3)=((2−2((1/2)))/3)=(1/3) •y=((2−2((2/5)))/3)=(6/(15))=(2/5) (x,y)={((1/2),(1/3)),((2/5),(2/5))}$
	$2 x + 3 y = 2 \Rightarrow y = \frac{2 - 2 x}{3}$ $x + y = 5 x y \Rightarrow x + \frac{2 - 2 x}{3} = 5 x (\frac{2 - 2 x}{3})$ $\Rightarrow 3 x + 2 - 2 x = 10 x - 10 x^{2}$ $\Rightarrow 10 x^{2} - 9 x + 2 = 0$ $10 x^{2} - 5 x - 4 x + 2 = 0$ $5 x (2 x - 1) - 2 (2 x - 1) = 0$ $(2 x - 1) (5 x - 2) = 0$ $x = \frac{1}{2} ∣ x = \frac{2}{5}$ $∙ y = \frac{2 - 2 x}{3} = \frac{2 - 2 (\frac{1}{2})}{3} = \frac{1}{3}$ $∙ y = \frac{2 - 2 (\frac{2}{5})}{3} = \frac{6}{15} = \frac{2}{5}$ $(x, y) = {(\frac{1}{2}, \frac{1}{3}), (\frac{2}{5}, \frac{2}{5})}$
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