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Question Number 81786 by M±th+et£s last updated on 15/Feb/20

Commented by M±th+et£s last updated on 16/Feb/20

$n o i t s r i g h t n o p r o b l e m w i t h t b e a n s$ $b u t i n e e d t o n o w h o w y o u g e t (x, y, z) = (1, 1, 1)$
Commented by john santu last updated on 15/Feb/20

$x + \frac{y}{z} = 2$ $y + \frac{z}{x} = 2$ $z + \frac{x}{y} = 2$ $\Rightarrow (x + y + z) + \frac{y}{z} + \frac{z}{x} + \frac{x}{y} = 6$ $(x + y + z) + \frac{z^{2} + x y}{x z} + \frac{x}{y} = 6$ $(x + y + z) + \frac{{y z}^{2} + {x y}^{2} + x^{2} z}{x y z} = 6 (*)$ $(x + \frac{y}{z}) (y + \frac{z}{x}) (z + \frac{x}{y}) = 8$ $(x y + z + \frac{y^{2}}{z} + \frac{y}{x}) (z + \frac{x}{y}) = 8$ $(x y z + x^{2} + z^{2} + \frac{x z}{y} + y^{2} + \frac{x y}{z} + \frac{y z}{x} + 1) = 8$ $x^{2} + y^{2} + z^{2} + x y z + \frac{x z}{y} + \frac{x y}{z} + \frac{y z}{x} = 7$ $x^{2} + y^{2} + z^{2} + x y z + \frac{{x z}^{2} + {x y}^{2}}{y z} + \frac{y z}{x} = 7$ $x^{2} + y^{2} + z^{2} + x y z + \frac{x^{2} z^{2} + x^{2} y^{2} + y^{2} z^{2}}{x y z} = 7$
Commented by john santu last updated on 15/Feb/20

$x = y = z = 1 \Rightarrow x + y + z = 3$
Commented by M±th+et£s last updated on 15/Feb/20

$t h a n k y o u s i r f o r g h i s s o l u t i o n . I w a n t$ $m e n t i o n t h a t t h e n u m b e r s a r e r e a l p o s a t i v e$ $h o w d i d y o u c o n c l u d e t h a t t h e o n l y o f$ $t h e r e n u m b e r s i s (1, 1, 1), I g u e s s t h a t$ $n e e d s s o m e p f o o f s . w e a r e i n t e r e s t e d$ $i n f i n d i n g t h e s u m i n s t e a d o f t h e r e$ $n u m b e r s v a l u e s .$
Commented by john santu last updated on 16/Feb/20

$m y a n s i s w r o n g ?$
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