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 Homework Statement:

The way we stipulate the conditions for the Gibbs Free Energy Equation (GFEE)  it seems to me that ΔS(system) must always equal negative ΔS(surroundings) ? Obviously this is incorrect but I can't see why. The GFEE says the magnitude of the entropy change of the surroundings is given by
ΔS(surr) = ΔH(syst)/T
Since the only exchange between the systemsurroundings is the heat of ΔH(syst), then the change of heat in the system is tautologically also the same ΔH that the surroundings experience. So then surely the magnitude of the entropy change of the system is also the same quantity as the above expression for that of the surroundings. Because both system and surroundings have the same magnitude of heat change at the same temperature with no PV work or matter exchange i.e. it erroneously appears to me that
ΔS(syst) = ΔH(syst)/T = ΔS(surr)
But of course if this were true it would render the GFEE useless since ΔG would always be zero.
So why does the normal entropy equation ΔS=q/T not work for the entropy change of the SYSTEM in the GFEE?
What instead is the expression (or way to consider) the entropy change of the system in the GFEE ?
I have never seen this discussed in my Thermodynamics text books or anywhere online.
 Relevant Equations:

ΔS(univ) = ΔS(surr) + ΔS(syst)
ΔS = q/T
q=ΔH
ΔS(surr) = ΔH(syst)/T
leading towards Gibbs:
ΔS(univ) = ΔH(syst)/T + ΔS(syst)
multiply by T to give Gibbs Free Energy Equation
T.ΔS(univ) = ΔG(univ) = ΔH(syst)  T.ΔS(syst)
What is the entropy change of the system in the Gibbs Free Energy Equation?
The general expression for entropy change is ΔS=q/T
The only exchange between the system and the surroundings is ΔH done reversibly, with no PV work and no matter transfer, therefore
q(syst) = ΔH(syst)
therefore surely the entropy change of the system is given by
ΔS(syst) = q(syst)/T
therefore
ΔS(syst) = ΔH(syst)/T
but I know this isn't correct, i just can't see why this expression for entropy change of the system is incorrect
The general expression for entropy change is ΔS=q/T
The only exchange between the system and the surroundings is ΔH done reversibly, with no PV work and no matter transfer, therefore
q(syst) = ΔH(syst)
therefore surely the entropy change of the system is given by
ΔS(syst) = q(syst)/T
therefore
ΔS(syst) = ΔH(syst)/T
but I know this isn't correct, i just can't see why this expression for entropy change of the system is incorrect