<s>[INST] Need an explanation of a particular expression transformation Please, I need an explanation of the one transformation. I have the equation set and its solution.
$$
\begin{cases}
    \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 3\\\\
    \frac{y}{x} + \frac{z}{y} + \frac{x}{z} = 3\\\\
    \ x + y + z = 3
\end{cases}
$$
In the solution three new variables were introduced:
$$
u = \frac{x}{y}; v = \frac{y}{z}; w = \frac{z}{x}
$$
And then using this new vars equation set became this:
$$
\begin{cases}
    \ u + v + w = 3\\\\
    \frac{1}{u} + \frac{1}{v} + \frac{1}{w} = 3\\\\\\
    \ uvw = 1
\end{cases}
$$
I can't understand how the $x + y + z = 3$ has become the $uvw = 1$. Can someone explain what have been done here?
My appreciation.
 [/INST] $$
uvw = \frac{x}{y}\frac{y}{z} \frac{z}{x}=\frac{xyz}{yzx}=1
$$
</s>