The principle of detailed balance for the generalized mass action law is: For given values k r there exists a positive equilibrium a i e q > 0 that satisfies detailed balance, that is, w r + = w r − − . This means that the system of linear detailed balance equations ∑ ∑ i γ γ r i x i = ln ⁡ ⁡ k r + − − ln ⁡ ⁡ k r − − = ln ⁡ ⁡ K r is solvable ( x i = ln ⁡ ⁡ a i e q ). The following classical result gives the necessary and sufficient conditions for the existence of a positive equilibrium a i e q > 0 with detailed balance (see, for example, the textbook).