Writing partial derivatives as a Jacobian matrix?

Question

In this pdf http://www.scs.illinois.edu/schulten/classes/Chem544/Notes/jacobian.pdf it states that we can write partial the partial derivative of the function B(x,y,z) with respect to z as shown in equation (2) in this document (I would write it out here but it seems a bit pointless). In this equation why does $\frac{\partial x}{\partial y}=0$ along with all the others they have called 0. Surly I can just rearrange B(x,y,z) to get a function (if possible) with the only $y$ one one side and only the other variables on the other and partially differentiation this and I would not get 0 as my answer. For example if we let: $$B=x+y+z$$ then $$y=B-x-z$$ and thus $$\frac{\partial x}{\partial y}=-1\ne0$$ I assume that it they make it out to be 0 since they are considering some variables constant, if this is the case how do we know which have been kept constant. If this is not the case please explain why they make it out to be 0?