Math Problem Statement
реши методом Гаусса
реши методом Гаусса
Solution
The system of linear equations is:
3x_1 + 2x_2 - 5x_3 + x_4 &= 14, \\ -4x_1 + 3x_2 + 4x_3 - x_4 &= 1, \\ 5x_1 + 5x_2 + 2x_3 + 3x_4 &= 18, \\ -2x_1 - 7x_2 + 6x_3 - x_4 &= -29. \end{aligned}$$ We will solve this using the **Gauss elimination method**, which involves forming the augmented matrix and reducing it to row echelon form to find the solution. Let me proceed with the calculations. numpy as np # Define the augmented matrix for the system of equations aug_matrix = np.array([ [3, 2, -5, 1, 14], [-4, 3, 4, -1, 1], [5, 5, 2, 3, 18], [-2, -7, 6, -1, -29] ], dtype=float) # Perform Gaussian elimination def gaussian_elimination(matrix): rows, cols = matrix.shape for i in range(rows): # Make the diagonal element 1 (if it's not already) if matrix[i, i] != 0: matrix[i] = matrix[i] / matrix[i, i] # Eliminate entries below the pivot for j in range(i + 1, rows): matrix[j] = matrix[j] - matrix[j, i] * matrix[i] # Back substitution to find the solution for i in range(rows - 1, -1, -1): for j in range(i - 1, -1, -1): matrix[j] = matrix[j] - matrix[j, i] * matrix[i] return matrix # Apply Gaussian elimination to the augmented matrix reduced_matrix = gaussian_elimination(aug_matrix.copy()) # Extract the solution (last column of the reduced matrix) solution = reduced_matrix[:, -1] reduced_matrix, solutionThe system of equations has been solved using Gaussian elimination. The solution is: $$\begin{aligned} x_1 &= -63, \\ x_2 &= -13, \\ x_3 &= -17, \\ x_4 &= 144. \end{aligned}$$ Let me know if you'd like a detailed step-by-step explanation of the Gaussian elimination process! ### Related Questions: 1. How does Gaussian elimination compare to other methods, such as Cramer's rule or LU decomposition? 2. Can Gaussian elimination handle inconsistent or dependent systems of equations? 3. What are the computational complexities of Gaussian elimination for large matrices? 4. How does partial or full pivoting improve the stability of Gaussian elimination? 5. Could this system be solved more efficiently using matrix algebra in Python? ### Tip: When solving systems of equations, always check for singular matrices (determinant = 0) to avoid division errors or inconsistent solutions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Formulas
Row operations for Gaussian elimination
Theorems
Fundamental Theorem of Linear Algebra
Suitable Grade Level
Grades 10-12 or College Level