Math Problem Statement
Solve the following system of equations by using Gauss-Seidal method. 28x + 4y - z = 32, x + 3y + 10z = 24, 2x + 17y + 4z = 35
Solution
The Gauss-Seidel method is an iterative technique for solving a system of linear equations. We need to first rewrite the given system of equations in a suitable form, where each equation is solved for one variable in terms of the others.
The system is:
28x + 4y - z &= 32 \quad \text{(1)} \\ x + 3y + 10z &= 24 \quad \text{(2)} \\ 2x + 17y + 4z &= 35 \quad \text{(3)} \end{aligned}$$ We solve each equation for one variable: 1. From equation (1), solve for $$x$$: $$x = \frac{32 - 4y + z}{28}$$ 2. From equation (2), solve for $$y$$: $$y = \frac{24 - x - 10z}{3}$$ 3. From equation (3), solve for $$z$$: $$z = \frac{35 - 2x - 17y}{4}$$ ### Iterative Procedure: - Start with initial guesses for $$x_0$$, $$y_0$$, and $$z_0$$. Let’s assume $$x_0 = 0$$, $$y_0 = 0$$, and $$z_0 = 0$$. - Apply the iterative formulas using these initial guesses to get better approximations. #### Iteration 1: Using the initial guesses $$x_0 = 0$$, $$y_0 = 0$$, and $$z_0 = 0$$: 1. $$x_1 = \frac{32 - 4(0) + 0}{28} = \frac{32}{28} = 1.1429$$ 2. $$y_1 = \frac{24 - 1.1429 - 10(0)}{3} = \frac{24 - 1.1429}{3} = \frac{22.8571}{3} = 7.6190$$ 3. $$z_1 = \frac{35 - 2(1.1429) - 17(7.6190)}{4} = \frac{35 - 2.2858 - 129.523}{4} = \frac{-96.8088}{4} = -24.2022$$ #### Iteration 2: Using the updated values $$x_1 = 1.1429$$, $$y_1 = 7.6190$$, and $$z_1 = -24.2022$$: 1. $$x_2 = \frac{32 - 4(7.6190) + (-24.2022)}{28} = \frac{32 - 30.476 + (-24.2022)}{28} = \frac{-22.6782}{28} = -0.8107$$ 2. $$y_2 = \frac{24 - (-0.8107) - 10(-24.2022)}{3} = \frac{24 + 0.8107 + 242.022}{3} = \frac{266.8327}{3} = 88.9442$$ 3. $$z_2 = \frac{35 - 2(-0.8107) - 17(88.9442)}{4} = \frac{35 + 1.6214 - 1512.0514}{4} = \frac{-1475.4300}{4} = -368.8575$$ The process would continue until the values of $$x$$, $$y$$, and $$z$$ converge to a stable solution. ### Tip: The Gauss-Seidel method converges faster if the system of equations is diagonally dominant, meaning that for each equation, the absolute value of the diagonal coefficient is greater than the sum of the absolute values of the other coefficients in that equation. Would you like to see further iterations or a stopping criterion explained? ### Related Questions: 1. How does the Gauss-Seidel method compare to the Jacobi method? 2. What are the conditions for convergence in the Gauss-Seidel method? 3. Can Gauss-Seidel be applied to non-diagonally dominant systems? 4. How does relaxation affect the convergence of the Gauss-Seidel method? 5. What are the advantages of using iterative methods over direct methods like Gaussian elimination?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Numerical Methods
Linear Algebra
Iterative Methods
Gauss-Seidel Method
Formulas
x = (32 - 4y + z) / 28
y = (24 - x - 10z) / 3
z = (35 - 2x - 17y) / 4
Theorems
Convergence of Iterative Methods
Diagonal Dominance
Suitable Grade Level
Undergraduate - Numerical Analysis/Linear Algebra
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