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You are asked to solve the nonlinear system of equations using the **Newton-Raphson Method**. The system is:

\[
\begin{aligned}
x^2 + y^2 &= 4 \\
x^2 + y^2 &= 8
\end{aligned}
\]

However, these equations seem identical except for the constants on the right-hand side. This would generally be inconsistent because two equations that differ only by constants cannot share common solutions. There might be a mistake in copying or formulating the problem.

Let me explain the general approach to solve a nonlinear system using Newton-Raphson.

### Steps for Newton-Raphson Method:
1. **Define the system of equations** as \( F_1(x, y) \) and \( F_2(x, y) \).
   
   In a general case, assume:
   \[
   F_1(x, y) = f_1(x, y) = 0
   \]
   \[
   F_2(x, y) = f_2(x, y) = 0
   \]

2. **Initial guess**: Choose an initial approximation \((x_0, y_0)\).

3. **Jacobian matrix**: Compute the Jacobian matrix \( J(x, y) \), which is the matrix of partial derivatives of the system:
   \[
   J(x, y) = \begin{bmatrix}
   \frac{\partial F_1}{\partial x} & \frac{\partial F_1}{\partial y} \\
   \frac{\partial F_2}{\partial x} & \frac{\partial F_2}{\partial y}
   \end{bmatrix}
   \]

4. **Iteration update**: Use the formula
   \[
   \begin{bmatrix}
   \Delta x \\
   \Delta y
   \end{bmatrix}
   = - J^{-1}(x_n, y_n) \begin{bmatrix}
   F_1(x_n, y_n) \\
   F_2(x_n, y_n)
   \end{bmatrix}
   \]
   and update the guess:
   \[
   \begin{bmatrix}
   x_{n+1} \\
   y_{n+1}
   \end{bmatrix}
   = \begin{bmatrix}
   x_n \\
   y_n
   \end{bmatrix}
   + \begin{bmatrix}
   \Delta x \\
   \Delta y
   \end{bmatrix}
   \]

5. **Convergence**: Repeat until the values converge within a desired tolerance.

Please confirm if the system is correctly written, or provide more details if the equations should be different. 

Would you like me to proceed with an example of the Newton-Raphson method based on a more consistent system? Or clarify the problem further?

### Relative Questions:
1. Do you want to modify the system of equations for the Newton-Raphson method?
2. Are you familiar with how the Jacobian matrix is derived for a nonlinear system?
3. Would you like to go through an example of the Newton-Raphson process for two variables?
4. Should we explore the convergence conditions of Newton-Raphson in more depth?
5. Would you like me to explain how to set up initial guesses in such problems?

### Tip:
Newton-Raphson is sensitive to the initial guess. A poor initial guess can lead to divergence or slow convergence.

Find the root of the nonlinear system using the Newton-Raphson method:

x^2 + y^2 = 4
x^2 + y^2 = 8

Learn how to apply the Newton-Raphson method to solve a nonlinear system of equations. This guide covers the numerical steps, Jacobian matrix computation, and iteration process. Suitable for undergraduate students studying numerical methods.

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