x-xy-y-14-and-x-2-xy-y-2-84-find-x-and-y-

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 + \sqrt{x y} + y = 14 \dots (i)

x^{2} + x y + y^{2} = 84 \dots (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

{\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

\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)}