System of non linear equations, find a good initial guess

Question

I don't know if this is the right section to ask this question. I have to find the solutions of: \begin{align} & k_r^*=\sum_{c=1}^{N_C} \frac{x_ry_c}{1+x_ry_c} \ \ \ r=1,...,N_R\\&h_c^*=\sum_{r=1}^{N_R} \frac{x_ry_c}{1+x_ry_c} \ \ \ c=1,...,N_C \end{align} where $ k_r^*,h_c^*,N_R,N_C$ are fixed, and $ x_r,y_c$ are positive. I'm trying to solve these with a nonlinear system solver, but it tells me "the Jacobian is too ill conditioned". I should try to find a good initial guess to pass to the solver, so I'm trying to do some analytic consideration on the system.