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Let \(x, y,\) and \(z\) be nonzero real numbers, such that no two are equal, and

\(\begin{align} x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x} \end{align}\)

 

Find all possible numeric values of \(xyz \).

 

 

Thanks in advance! (also, as a bonus question, how do you center \(\LaTeX\) in web2.0calc? Thanks. 

 May 26, 2021
 #1
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I think it is double dollar signs:

 

$$0$$

 

Hint: Multiply the equations by xyz.

 May 26, 2021
 #2
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$$ wow this works thanks. $$

 

Thank you! Will try the hint. 

xCorrosive  May 26, 2021
 #3
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No problem! Also, if you want LaTeX text, you can use the \text{} environment:

 

$$\text{Wow, this is clearer text!}$$

MathProblemSolver101  May 26, 2021
 #4
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$$ \text{cool!} $$

 

Anyway, I didn't exactly multiply by $xyz$, but now I'm at a point where I have this equation

 

$(y-x)(y-z)(z-x)= \frac{(y-x)(y-z)(z-x)}{x^2y^2z^2}$

 

Any pointers on how to continue from here? 

xCorrosive  May 26, 2021
 #5
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Wait I have a thought. 

 

There are only two cases in which these two equations are equal to each other.

 

The first case is when two the variables are $0$, effectively leading the equation to be zero. 

 

The second case is when $x^2y^2z^2 = 1$. 

 

We're looking for the square root of this, so does it mean that the only solutions are $-1$ and $1$. Or are there other possible solutions? 

xCorrosive  May 26, 2021

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