So I am having real trouble understanding what equations of state are and how we form them. My issue stems from reading multiple sources. So I understand that an equation of state is used to build a relationship between variables to describe a state of a system.
For example $P=P(V,T)$, where $P=NKT/V$, is a function of state, but then I started reading about Gibbs, Helmholtz, enthalpy, etc and suddenly am very confused.
So looking at Gibbs, for example, we have two equations $$G=E+PV-TS$$ and $$dG=-SdT+VdP$$ but both describe the state of the system. I did not think an equation of state could be a differential equation but according to one book I have read, $dE$ is related to equation of state and this has caused me great confusion.