Solution to system of the form A(y)X=B

Question

I have a system of non-equilibrium equations of the form

$A(y)X=B \quad\quad\quad$ Eq(1)

$X_1=exp(\Sigma_{i=2}^{N}ln(X_i)) \quad\quad\quad$ Eq(2)

Now I have N+1 equations and N+1 unknowns ($X_1$ to $X_N$ & y).

The way I am going about solving this is

1>Guess Value of y
2>Calculate X_1 to X_N from eqn1
3>Calculate new X_1 value from eqn2
4>Use the new X_1 and previous X_2:X_N values to find y from eqn1
5>Iterate from step <2> until y_old-y_new < error

The initial guess of y is very critical for convergence and I am not able to generalize the solver. Please suggest if there is an issue with the algorithm or any other methods to solve this system

Thanks in advance

EDIT

Simple Case $$\begin{matrix} \left( \begin{array}{cc} c_1y & c_2 & c_3\\ c_3 & c_4y & c_6\\ c_7 & c_8 & c_9y \end{array} \right) % \left( \begin{array}{cc} X_1 \\ X_2 \\ X_3 \end{array} \right) % = % \left( \begin{array}{cc} B_1 \\ B_2 \\ B_3 \end{array} \right) % \quad\quad\quad Eq(1) \end{matrix}$$ $X_1=c_{10}exp(c_{11}ln(X_2)+c_{12}ln(X_3)) \quad\quad \quad\quad Eq(2)$

$c_1$ to $c_{12}$ are constants

$X_i \& y$ are unknowns