Given the equation $$ x+y+z=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} =1 $$ . Prove that at least one of $$ x, y, z $$ is equal to $$ 1 $$

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Given the equation $$ x+y+z=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} =1 $$ . Prove that at least one of $$ x, y, z $$  is equal to $$ 1 $$

Uniteasy Admin · Accepted Answer

Remove denominator from the fraction
$$\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} =1$$
$$ xy+yz+zx=xyz $$
Rearrange the terms
$$ xyz-xy-yz-zx+1-1=0 $$
Replace the first 1 with x+y+z based on given condition
$$ xyz-xy-yz-zx+x+y+z-1=0 $$
Factoring by grouping the terms 
$$ (x-1)(y-1)(z-1)=0 $$
Therefore, in order for the equation to be valid, at least one of the variables $$x, y, z$$ is equal to 1.

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