1 &1

If $xyz=1$
prove that:

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

#Algebra

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

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:

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

Alex Zhong - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$