Solve System of Linear Equations Using Matrix Methods

Math Problem Statement

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The system of equations extracted from the image is:

2x_1 + x_2 - x_3 = -1 \\ -x_1 - 2x_2 + 3x_3 = 1 \\ 4x_1 + 3x_2 - 5x_3 = -1 \end{cases}$$ We will solve this system using matrix methods (Gaussian elimination or substitution). Let me compute it. numpy as np # Coefficient matrix and constants vector A = np.array([ [2, 1, -1], [-1, -2, 3], [4, 3, -5] ]) b = np.array([-1, 1, -1]) # Solve the system solution = np.linalg.solve(A, b) solutionThe solution to the system of equations is: $$x_1 = 0, \quad x_2 = -2, \quad x_3 = -1$$ Would you like me to explain the steps in more detail or verify this solution? ### Related Questions: 1. How can Gaussian elimination be used to solve this system step by step? 2. What are the properties of the coefficient matrix in this system (e.g., determinant, rank)? 3. How can this system be solved graphically in 3D? 4. How would the solution change if one equation were modified slightly? 5. Can systems of this type always be solved using matrix methods? ### Tip: When solving systems of equations, always check the determinant of the coefficient matrix to confirm that a unique solution exists.

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Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Matrix Methods

Matrix equation Ax = b
Gaussian elimination steps for solving linear systems

Properties of matrices for linear systems
Uniqueness of solutions based on the determinant

Grades 11-12 and undergraduate level

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