{ (((√(x/y))−(√(y/x))=(3/2))),((x+y+xy=9)) :}


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Question Number 165881 by bounhome last updated on 09/Feb/22

${\begin{cases} \sqrt{\frac{x}{y}} - \sqrt{\frac{y}{x}} = \frac{3}{2} \\ x + y + x y = 9 \end{cases}$
	Answered by Rasheed.Sindhi last updated on 09/Feb/22
	${ (((√(x/y))−(√(y/x))=(3/2))),((x+y+xy=9)) :} (√(x/y)) =a: a−(1/a)=(3/2) 2a^2 −3a−2=0 (a−2)(2a+1)=0 a=2 ∣ a=−(1/2) (√(x/y)) =2 ∣ (√(x/y)) =−(1/2) (Rejected) (x/y)=4⇒x=4y x+y+xy=9⇒4y+y+4y×y=9 4y^2 +5y−9=0 (y−1)(4y+9)=0 y=1 ∣ y=−(9/4) x=4 ∣ x=−9 S_1 ={(4,1),(−9,−(9/4))}$
	${\begin{cases} \sqrt{\frac{x}{y}} - \sqrt{\frac{y}{x}} = \frac{3}{2} \\ x + y + x y = 9 \end{cases}$ $\sqrt{\frac{x}{y}} = a :$ $a - \frac{1}{a} = \frac{3}{2}$ $2 a^{2} - 3 a - 2 = 0$ $(a - 2) (2 a + 1) = 0$ $a = 2 ∣ a = - \frac{1}{2}$ $\sqrt{\frac{x}{y}} = 2 ∣ \sqrt{\frac{x}{y}} = - \frac{1}{2} (R e j e c t e d)$ $\frac{x}{y} = 4 \Rightarrow x = 4 y$ $x + y + x y = 9 \Rightarrow 4 y + y + 4 y \times y = 9$ $4 y^{2} + 5 y - 9 = 0$ $(y - 1) (4 y + 9) = 0$ $y = 1 ∣ y = - \frac{9}{4}$ $x = 4 ∣ x = - 9$ $S_{1} = {(4, 1), (- 9, - \frac{9}{4})}$
	Answered by MJS_new last updated on 09/Feb/22

	$let t = \sqrt{\frac{x}{y}} \land t ⩾ 0$ $t - \frac{1}{t} = \frac{3}{2} \Rightarrow t = 2$ $\sqrt{\frac{x}{y}} = 2 \Rightarrow y = \frac{x}{4}$ $x + \frac{x}{4} + x \times \frac{x}{4} = 9 \Leftrightarrow x^{2} + 5 x - 36 = 0$ $\Rightarrow x = - 9 \land y = - \frac{9}{4} \lor x = 4 \land y = 1$
	Answered by Rasheed.Sindhi last updated on 10/Feb/22
	${ (((√(x/y))−(√(y/x))=(3/2))),((x+y+xy=9)) :} An Alternate { ((((x−y)/( (√(xy))))=(3/2).........(i))),((x+y=9−xy.....(ii))) :} ⇒ { (((x−y)^2 =(((3(√(xy)) )/2))^2 ......(iii))),(((x+y)^2 =(9−xy)^2 .....(iv))) :} (iv)−(iii):4xy=(9−xy)^2 −(((3(√(xy)) )/2))^2 4xy=81−18xy+(xy)^2 −((9xy)/4) 4(xy)^2 −72xy−9xy−16xy+324=0 4(xy)^2 −97xy+324=0 xy=((−(−97)±(√((−97)^2 −4∙4∙324)))/(2∙4)) xy=((97±65)/8)=((162)/8) , ((32)/8)=((81)/4) , 4 xy=81/4 : (i): ((x−y)/( (√((81)/4))))=(3/2)⇒x−y=((27)/4) (ii): x+y=9−((81)/4)=−((45)/4) x=−(9/4),y=−9 xy=4 : (i): ((x−y)/( (√4)))=(3/2)⇒x−y=3 (ii): x+y=9−4=5 x=4,y=1$
	${\begin{cases} \sqrt{\frac{x}{y}} - \sqrt{\frac{y}{x}} = \frac{3}{2} \\ x + y + x y = 9 \end{cases}$ $A n A lternate$ ${\begin{cases} \frac{x - y}{\sqrt{x y}} = \frac{3}{2} . . . . . . . . . (i) \\ x + y = 9 - x y . . . . . (i i) \end{cases}$ $\Rightarrow {\begin{cases} {(x - y)}^{2} = {(\frac{3 \sqrt{x y}}{2})}^{2} . . . . . . (i i i) \\ {(x + y)}^{2} = {(9 - x y)}^{2} . . . . . (i v) \end{cases}$ $(i v) - (i i i) : 4 x y = {(9 - x y)}^{2} - {(\frac{3 \sqrt{x y}}{2})}^{2}$ $4 x y = 81 - 18 x y + {(x y)}^{2} - \frac{9 x y}{4}$ $4 {(x y)}^{2} - 72 x y - 9 x y - 16 x y + 324 = 0$ $4 {(x y)}^{2} - 97 x y + 324 = 0$ $x y = \frac{- (- 97) \pm \sqrt{{(- 97)}^{2} - 4 \cdot 4 \cdot 324}}{2 \cdot 4}$ $x y = \frac{97 \pm 65}{8} = \frac{162}{8}, \frac{32}{8} = \frac{81}{4}, 4$ $\underset{―}{x y = 81 / 4 :}$ $(i) : \frac{x - y}{\sqrt{\frac{81}{4}}} = \frac{3}{2} \Rightarrow x - y = \frac{27}{4}$ $(i i) : x + y = 9 - \frac{81}{4} = - \frac{45}{4}$ $x = - \frac{9}{4}, y = - 9$ $\underset{―}{x y = 4 :}$ $(i) : \frac{x - y}{\sqrt{4}} = \frac{3}{2} \Rightarrow x - y = 3$ $(i i) : x + y = 9 - 4 = 5$ $x = 4, y = 1$
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