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.