I am reading a math review in my thermodynamics text and I a little confused by this. Here is the excerpt:

I am confused by the part where it says

If they selected x = (y, z) then isn't that saying that x is dependent on y? So how can we just turn around and say y = y(x, z) ? That is, if we selected x as dependent in the first function, why can we turn around and call it independent in the second.

Sorry, this might be a stupid question. I just don't see why we bother calling variables independent and dependent in a situation like this?

As a simple example if you had the equation x+y+z=0 you could write any variable as a function of the other two quite simply.

The dependent/independent lines are obviously blurred here; you just use them for the purposes of being able to describe what counts as a function and what's being considered as a variable when differentiating