I am trying to find numerically $\arg\min_{x\in(1, +\infty),y\in(0, 1)}\sum_i\log(xy+\alpha_ix+\beta_iy+\gamma_i)$, where the sum has a large number of terms, and the coefficients are such that the expression is always defined ($\alpha_i, \gamma_i$>0, $\beta_i>-1$).

Of course, one can rephrase this as trying to solve the following system: $\sum_i\frac{y+\alpha_i}{xy+\alpha_ix+\beta_iy+\gamma_i}=\sum_i\frac{x+\beta_iy}{xy+\alpha_ix+\beta_iy+\gamma_i}=0$.

Is there any theory behind this kind of equations? (an actual way to solve this is welcome but I'd like to know about theory as well.)

Thanks in advance.

Edit: more precise wording.