Math Problem Statement
solve the system 2x-4y+5z=7, -3x-2y+2z=10,-2x+3y-2z=14
Solution
We are solving the system of linear equations:
Solution
We will use the matrix method to find the solution, representing the system in the form , where:
2 & -4 & 5 \\ -3 & -2 & 2 \\ -2 & 3 & -2 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 7 \\ 10 \\ 14 \end{bmatrix}.$$ To solve, $$\mathbf{x} = A^{-1}\mathbf{b}$$. #### Step 1: Compute $$A^{-1}$$ The inverse of $$A$$ is computed as $$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$$, where $$\det(A)$$ is the determinant of $$A$$, and $$\text{adj}(A)$$ is the adjugate matrix. #### Step 2: Solve for $$\mathbf{x}$$ Perform $$A^{-1} \mathbf{b}$$ to find the solution. I'll compute the values explicitly. numpy as np # Define the coefficient matrix A and the constant vector b A = np.array([[2, -4, 5], [-3, -2, 2], [-2, 3, -2]]) b = np.array([7, 10, 14]) # Solve the system using matrix inversion solution = np.linalg.solve(A, b) solutionThe solution to the system of equations is: $$x = -2.90, \quad y = 9.52, \quad z = 10.17$$ ### Final Answer: - $$x \approx -2.90$$ - $$y \approx 9.52$$ - $$z \approx 10.17$$ Would you like a detailed explanation of the steps, or do you have any questions? --- ### Related Questions: 1. How can the determinant of a matrix help identify if a system has a unique solution? 2. What are the advantages of using matrix methods over substitution or elimination methods? 3. How does changing one equation affect the solution of the system? 4. Can you use the Cramer's Rule to solve this system? 5. What are the computational complexities of solving linear systems using matrices? --- ### Tip: For large systems, numerical methods like LU decomposition or iterative methods (e.g., Jacobi or Gauss-Seidel) are more efficient than direct matrix inversion.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Operations
Formulas
Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix
Matrix inversion: A^(-1) = adj(A)/det(A)
Matrix solution: x = A^(-1)b
Theorems
Matrix Inversion Theorem
Properties of Determinants
Existence and Uniqueness Theorem for Linear Systems
Suitable Grade Level
Grades 10-12 and Undergraduate
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