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# Algebra Question - Please help quickly.

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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

### 5+0 Answers

#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
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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
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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