Solve for x and y x (√x) + y(√y) = 182 ..... (i) x (√y) + y(√x) = 183 ..... (ii)


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Question Number 56321 by Tawa1 last updated on 14/Mar/19

$Solve for x and y$ $x \sqrt{x} + y \sqrt{y} = 182 . . . . . (i)$ $x \sqrt{y} + y \sqrt{x} = 183 . . . . . (ii)$
	Answered by tanmay.chaudhury50@gmail.com last updated on 14/Mar/19

	$x = a^{2} y = b^{2}$ $a^{3} + b^{3} = 182$ $a^{2} b + {a b}^{2} = 183$ ${(a + b)}^{3} - 3 a b (a + b) = 182$ ${(a + b)}^{3} - 3 \times 183 = 182$ ${(a + b)}^{3} = 182 + 549 = 731$ $(a + b) = {(731)}^{\frac{1}{3}} \approx 9.008 \to c o n s i d e r 9$ $a + b = 9$ $a b (a + b) = 183$ $a b = \frac{183}{{(731)}^{\frac{1}{3}}} = \frac{183}{9.008} = 20.32$ $f o r c a l c u l a t i o n e a s y . . a + b = 9$ $a b = 20$ $(9 - b) b = 20$ $9 b - b^{2} - 20 = 0$ $b^{2} - 9 b + 20 = 0$ $(b - 4) (b - 5) = 0$ $b = 4 a n d 5$ $a = 5 a n d 4$ $x = 25 a n d 16$ $y = 16 a n d 25$
	Commented by tanmay.chaudhury50@gmail.com last updated on 14/Mar/19

	$t h e g i v e n p r o b l e m c a n b e s o l v e d b u t i t n e e d c a l c u l a t i o n$ $s o i a p p r o x i m a t e d i t . . .$
	Commented by Tawa1 last updated on 14/Mar/19

	$God bless you sir$
	Answered by behi83417@gmail.com last updated on 14/Mar/19
	$x=t^2 ,y=s^2 ⇒ { ((t^3 +s^3 =182)),((t^2 s+s^2 t=183)) :}⇒_(ts=q) ^(t+s=p) { (((t+s)(t^2 +s^2 −ts)=182)),((ts(t+s)=183)) :} ⇒ { ((p(p^2 −3q)=182)),((qp=183)) :}⇒ { ((p^3 −3pq=182)),((pq=183)) :} ⇒p^3 =3×183+182=731 ⇒p#9⇒q#20 ⇒ { ((t+s=9)),((ts=20)) :}⇒ { ((t=4)),((s=5)) :}⇒ { ((x=16)),((y=25)) :} .■$
	$x = t^{2}, y = s^{2}$ $\Rightarrow {\begin{cases} t^{3} + s^{3} = 182 \\ t^{2} s + s^{2} t = 183 \end{cases} \underset{t s = q}{\overset{t + s = p}{\Rightarrow}} {\begin{cases} (t + s) (t^{2} + s^{2} - t s) = 182 \\ t s (t + s) = 183 \end{cases}$ $\Rightarrow {\begin{cases} p (p^{2} - 3 q) = 182 \\ q p = 183 \end{cases} \Rightarrow {\begin{cases} p^{3} - 3 p q = 182 \\ p q = 183 \end{cases}$ $\Rightarrow p^{3} = 3 \times 183 + 182 = 731$ $You can't use 'macro parameter character #' in math mode$ $\Rightarrow {\begin{cases} t + s = 9 \\ t s = 20 \end{cases} \Rightarrow {\begin{cases} t = 4 \\ s = 5 \end{cases} \Rightarrow {\begin{cases} x = 16 \\ y = 25 \end{cases} . ◼$
	Commented by Tawa1 last updated on 14/Mar/19

	$God bless you sir$
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