Constrained optimization with a system of nonlinear equations constraints

My optimization problem has 2 decision variables that I want to find the optimal solution to minimize an objective function. The constraint for the problem is a system of 11 equations with 11 unknowns (excluding the two decision variables). It is like if I know the value of the two decision variables, I will be able to solve for the 11 unknowns (my objective function is 1 of these 11 unknowns). So how do I approach to solve this kind of problem? One way I can think of is to rearrange the 11 equations-11 unknowns down to 2 equations written in terms of the two decision variables (but it seems too much work because not all of the 11 equations are linear). To be short, the problem is something like this

11 unknowns: nC_5; nH_5; nC_6; nH_6; n5; n6; T; P; H; L; V;

2 decision variables: n2 and n4

objective function: minimize nC_6 or nC_6/n6

subject to: 11 equations written in terms of 11 unknowns and 2 decision variables

[Walter Roberson]

I am not clear whether the value of the decision variables will be fixed for any one optimization, or if you want to find the four solution sets (each variable on or off), or if you want to optimize a set of 13 variables two of which are effectively binary?