x+(√(xy))+y=14 and x^2 +xy+y^2 =84 , find x and y


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Question Number 143328 by Rankut last updated on 13/Jun/21

$x + \sqrt{x y} + y = 14 a n d$ $x^{2} + x y + y^{2} = 84,$ $f i n d x a n d y$
	Answered by Rasheed.Sindhi last updated on 13/Jun/21
	$x+(√(xy))+y=14...(i) x^2 +xy+y^2 =84...(ii) (x+(√(xy))+y)^2 =14^2 x^2 +xy+y^2 +2x(√(xy))+2y(√(xy))+2xy=196 84+2x(√(xy))+2y(√(xy))+2xy=196 x(√(xy))+y(√(xy))+xy=56 (√(xy))(x+y+(√(xy)))=56 (√(xy))(14)=56 (√(xy))=4 xy=16⇒y=((16)/x) (i)⇒x+y=10⇒x+((16)/x)=10 x^2 −10x+16=0 (x−8)(x−2)=0 x=8,2⇒y=2,8 (x,y)={(8,2),(2,8)}$
	$x + \sqrt{x y} + y = 14 . . . (i)$ $x^{2} + x y + y^{2} = 84 . . . (i i)$ ${(x + \sqrt{x y} + y)}^{2} = 14^{2}$ $x^{2} + x y + y^{2} + 2 x \sqrt{x y} + 2 y \sqrt{x y} + 2 x y = 196$ $84 + 2 x \sqrt{x y} + 2 y \sqrt{x y} + 2 x y = 196$ $x \sqrt{x y} + y \sqrt{x y} + x y = 56$ $\sqrt{x y} (x + y + \sqrt{x y}) = 56$ $\sqrt{x y} (14) = 56$ $\sqrt{x y} = 4$ $x y = 16 \Rightarrow y = \frac{16}{x}$ $(i) \Rightarrow x + y = 10 \Rightarrow x + \frac{16}{x} = 10$ $x^{2} - 10 x + 16 = 0$ $(x - 8) (x - 2) = 0$ $x = 8, 2 \Rightarrow y = 2, 8$ $(x, y) = {(8, 2), (2, 8)}$
	Answered by Rasheed.Sindhi last updated on 13/Jun/21
	${ ((x+(√(xy))+y=14⇒(x+y)^2 =(14−(√(xy)))^2 )),((x^2 +xy+y^2 =84⇒(x+y)^2 =84+xy)) :} ⇒(14−(√(xy)))^2 =84+xy 196+xy^(×) −28(√(xy))=84+xy^(×) 28(√(xy))=196−84=112 xy=16 x+(√(xy))+y=14⇒x+y=10 (x,y)={(2,8),(8,2)}$
	${\begin{cases} x + \sqrt{x y} + y = 14 \Rightarrow {(x + y)}^{2} = {(14 - \sqrt{x y})}^{2} \\ x^{2} + x y + y^{2} = 84 \Rightarrow {(x + y)}^{2} = 84 + x y \end{cases}$ $\Rightarrow {(14 - \sqrt{x y})}^{2} = 84 + x y$ $196 + \overset{\times}{x y} - 28 \sqrt{x y} = 84 + \overset{\times}{x y}$ $28 \sqrt{x y} = 196 - 84 = 112$ $x y = 16$ $x + \sqrt{x y} + y = 14 \Rightarrow x + y = 10$ $(x, y) = {(2, 8), (8, 2)}$
	Answered by Rasheed.Sindhi last updated on 13/Jun/21
	$x+(√(xy))+y=14^((i)) ∧ x^2 +xy+y^2 =84^((ii)) (i)⇒xy=(14−x−y)^2 (ii)⇒xy=84−x^2 −y^2 (i)&(ii)⇒(14−x−y)^2 =84−x^2 −y^2 196+x^2 +y^2 −28x+2xy−28y=84−x^2 −y^2 2x^2 +2y^2 −28x+2xy−28y=84−196 x^2 +y^2 −14x+xy−14y=−56 84−xy−14x+xy−14y=−56 −14x−14y=−56−84=−140 x+y=10 (i):x+(√(xy))+y=14 ⇒(√(xy))=14−10=4 xy=16 x+y=10∧xy=16⇒(x,y)={(2,8),(8,2)}$
	$\overset{(i)}{x + \sqrt{x y} + y = 14} \land \overset{(i i)}{x^{2} + x y + y^{2} = 84}$ $(i) \Rightarrow x y = {(14 - x - y)}^{2}$ $(i i) \Rightarrow x y = 84 - x^{2} - y^{2}$ $(i) & (i i) \Rightarrow {(14 - x - y)}^{2} = 84 - x^{2} - y^{2}$ $196 + x^{2} + y^{2} - 28 x + 2 x y - 28 y = 84 - x^{2} - y^{2}$ $2 x^{2} + 2 y^{2} - 28 x + 2 x y - 28 y = 84 - 196$ $x^{2} + y^{2} - 14 x + x y - 14 y = - 56$ $84 - x y - 14 x + x y - 14 y = - 56$ $- 14 x - 14 y = - 56 - 84 = - 140$ $x + y = 10$ $(i) : x + \sqrt{x y} + y = 14 \Rightarrow \sqrt{x y} = 14 - 10 = 4$ $x y = 16$ $x + y = 10 \land x y = 16 \Rightarrow (x, y) = {(2, 8), (8, 2)}$
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