Solve-for-real-numbers-x-y-5-x-y-5-5-y-x-y-5-5-y-5-x-

Question Number 161744 by HongKing last updated on 21/Dec/21

Solve for real numbers :

\frac{x}{y} + \frac{5}{x} + \frac{y - 5}{5} = \frac{y + x}{y + 5} + \frac{5 + y}{5 + x}

Commented by Rasheed.Sindhi last updated on 22/Dec/21

N o t u n i q u e s o l u t i o n . B e c a u s e

o n l y o n e e q u a t i o n i s g i v e n i n t w o

v a r i a b l e s .

Answered by Rasheed.Sindhi last updated on 22/Dec/21

y = 5 :

\frac{x}{5} + \frac{5}{x} = \frac{5 + x}{10} + \frac{10}{5 + x}

\frac{x^{2} + 5^{2}}{5 x} = \frac{{(5 + x)}^{2} + 10^{2}}{10 (5 + x)}

\frac{x^{2} + 25}{x} = \frac{25 + 10 x + x^{2} + 100}{2 (5 + x)}

(x^{2} + 25) (2 x + 10) = x^{3} + {10 x}^{2} + 125 x

{2 x}^{3} + {10 x}^{2} + 50 x + 250 - x^{3} - {10 x}^{2} - 125 x

x^{3} - 75 x + 250 = 0

x = 5, - 10

(5, 5), (- 10, 5) a r e t w o o f m a n y

s o l u t i o n s .

Commented by HongKing last updated on 25/Dec/21

thank you so much my dear Sir