Is there a simple way of solving system of nonlinear equations given by
$ \sum_{i=1}^n \frac{a_{ij} x_j}{1+\sum_{k=1}^J a_{ik}x_k} = c_j $
where $a_{ij}$ and $c_j$ are constants. In other words, is there a change of variable method that makes solving this system numerically tractable than current specification?
These equations in rational functions correspond to a system of polynomial equations, which in general will "reduce" to a hellishly complicated polynomial in one variable. So you're not likely to get anywhere looking for closed-form solutions. Have you tried multivariate Newton-type numerical methods with the system as given?