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1 &1

If $xyz=1$
prove that:

$\frac { x }{ xy+x+1 } +\frac { y }{ yz+y+1 } +\frac { z }{ zx+z+1 } =1$

2 years, 1 month ago

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Substitute $$x=\dfrac{1}{yz}$$ into the first term of the expression:

$$\dfrac{\dfrac{1}{yz}}{\dfrac{y}{yz} + \dfrac{1}{yz}+ 1} = \dfrac{1}{yz+y + 1} .$$

Again, substitute $$x=\dfrac{1}{yz}$$ into the third term of the expression:

$$\dfrac{z}{\dfrac{z}{yz} + z+ 1} = \dfrac{yz}{yz+y + 1} .$$

Put the terms together we have:

$$\dfrac{1 + y + yz}{yz+y+1} = 1. \blacksquare$$ · 2 years, 1 month ago