# The Fixed Point Method for Solving Systems of Two Nonlinear Equations

We will now look at an extension to The Fixed Point Method for Approximating Roots. Suppose that a solution $(\alpha, \beta)$ exists to the system of two nonlinear equations:

(1)

We first rewrite this system of linear equations in the following form:

(2)

Let $(x_0, y_0)$ be an initial solution to this system that is sufficiently close to $(\alpha, \beta)$. Then for $n = 0, 1, 2, …$ define successive iterated approximations to the solution $(\alpha, \beta)$ with the formulas:

(3)

On the Convergence of The Fixed Point Method for Solving Systems of Two Nonlinear Equations we will provide criterion for which this method is guaranteed to converge to the solution $(\alpha, \beta)$ provided that the initial approximation $(x_0, y_0)$ meets certain requirements.