Math in Melitz and Ottaviano (2008)

Question

I am reading Melitz and Ottaviano (2008), but I find it hard to understand some math in the model.

The preference is given by :

, which implies a demand function:

Then the paper says the "inverse demand function can then be inverted to yield the linear market demand system for these varieties":

But what operations do they take in getting the function $q_i$?

I am very confused on dealing with the $Q_c$ variable, which is defined to be :

Thanks.

Answer 1

hint: sum equation (2) for all i and use the definition of $Q^c$, we get an expression for $Q^c$. then, substitute this expression into (2) and solve for $q^c$.