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Find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right. $$

Things I have done: first I show that all of these numbers should be bigger than $0$. If we take $x<0$ then this would led us to $z>3$ and $\frac{2}{y}>3$. By this we got $y+\frac{2}{z} < \frac{4}{3}$ which is false.summing these three equation and using AM-GM gives: $$9=x+\frac{1}{y}+\frac{1}{y}+y+\frac{1}{z}+\frac{1}{z}+z+\frac{1}{x}+\frac{1}{x}\ge9\sqrt[9]{\frac{1}{xyz}} \rightarrow xyz\ge1$$

so as $xyz\ge1$ we can conclude that $x\ge1,y\ge1,z\ge1$ because $x<1$ would led us to $y<1 $ and $z<1$ which makes $xyz<1$. I tried to show that $x>1,y>1,z>1$ leads to contradiction but I was not successful(well I was wrong as $x=y=z=2$ is an answer). Any other approaches on solving this or help on continuing my solution is appreciated.

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1 Answer 1

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We have $z=3-\frac{2}{x}$, so

$$ y=3-\frac{2}{z}=3-\frac{2}{3-\frac{2}{x}}= 3-\frac{2x}{3x-2}=\frac{7x-6}{3x-2} \tag{1} $$

Then

$$ 0=x+\frac{2}{y}-3=x+\frac{2(3x-2)}{7x-6}-3= \frac{7x^2 - 21x + 14}{7x-6} $$

We deduce $7x^2 - 21x + 14=0$, i.e. $x^2-3x+2=0$, or $(x-1)(x-2)=0$, so that $x=1$ or $x=2$.

In the end we have two solutions : $x=y=z=1$ or $x=y=z=2$.

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