Help Wanted!

Recently, while reading a book, I read that $xyz = x+y+z+2$ is equivalent to $\dfrac{1}{1+x} +\dfrac{1}{1+y} + \dfrac{1}{1+z}=1$ . The author implied that this was intuitive, but I cannot see where it comes from. Anyone like to clarify?

Easy Math Editor

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Comments

Let $a = 1 + x, b = 1 + y$ and $c = 1 + z$ . Then the first equation can be written as

$(a - 1)(b - 1)(c - 1) = a + b + c - 1 \Longrightarrow abc + (a + b + c) - (ab + ac + bc) - 1 = a + b + c - 1$

$\Longrightarrow abc = ab + ac + ab \Longrightarrow 1 = \dfrac{1}{c} + \dfrac{1}{b} + \dfrac{1}{a}.$

Now rearrange and re-substitute to end up with $\dfrac{1}{1 + x} + \dfrac{1}{1 + y} + \dfrac{1}{1 + z} = 1$ as desired.

This doesn't seem that "intuitive" to me. :)

Brian Charlesworth - 6 years, 2 months ago

Awesome, thanks!

Ryan Tamburrino - 6 years, 2 months ago

You're welcome. :)

Brian Charlesworth - 6 years, 2 months ago

Just in case anyone is curious as to what this was getting at, the author then went on and showed how the equation $xyz=x+y+z+2$ implies the existence of reals $a,b,c$ such that $x=\dfrac{b+c}{a}, y=\dfrac{a+c}{b}, z=\dfrac{a+b}{c}$ . All of which I did, in fact, understand. :)

Ryan Tamburrino - 6 years, 2 months ago

So will you post a question using the above mentioned Identity ?

A Former Brilliant Member - 6 years, 2 months ago

Sure, I'll get on that soon!

Ryan Tamburrino - 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}`	$\boxed{123}$