Using the Gauss-Seidel method for 5 iterations, the approximate solution to the system of equations is:

$x_1 \approx 0.1861, \quad x_2 \approx 0.3312, \quad x_3 \approx -0.4227$

Would you like to see more details on how the iterations were performed or do you have any other questions?

Here are five related questions to explore further:

1. How does the Gauss-Seidel method differ from the Jacobi method?
2. What is the convergence criteria for the Gauss-Seidel method?
3. Can this system be solved using matrix inversion? How would the solution compare?
4. What are the advantages of iterative methods like Gauss-Seidel over direct methods (e.g., Gaussian elimination)?
5. How would increasing the number of iterations improve the accuracy of the solution?

Tip: The Gauss-Seidel method converges faster if the matrix is diagonally dominant or symmetric positive definite. Always check these properties before applying the method!