I have the conditions $$1 = e^{\alpha-1} \sum_{n=1}^M e^{\beta E_n + \gamma N_n} $$ $$\langle E \rangle = e^{\alpha-1} \sum_{n=1}^M E_n e^{\beta E_n + \gamma N_n} $$ $$\langle N \rangle = e^{\alpha-1} \sum_{n=1}^M N_n e^{\beta E_n + \gamma N_n} $$
and from this I want to determine what $\alpha, \beta, \gamma$ are in terms of all the other variables. Is this possible in this generality?
Edit: Since all other variables are positive, taking the log might help.