Solving a System of Three Linear Equations Using Matrix Methods

Math Problem Statement

2x − 5y + 4z = 3 5x + 2y + z = 0 2x + 3y − 2z = 3

You have a system of three linear equations:

2x - 5y + 4z &= 3 \\ 5x + 2y + z &= 0 \\ 2x + 3y - 2z &= 3 \end{aligned}$$ ### Solving the system using the elimination or matrix method: We can write this system as a matrix equation: $$\begin{bmatrix} 2 & -5 & 4 \\ 5 & 2 & 1 \\ 2 & 3 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix}$$ Now, let's solve for $$x, y, z$$. numpy as np # Coefficient matrix A = np.array([[2, -5, 4], [5, 2, 1], [2, 3, -2]]) # Constants matrix B = np.array([3, 0, 3]) # Solve the system solution = np.linalg.solve(A, B) solutionThe solution to the system is: $$x = 2, \quad y = -3, \quad z = -4$$ Would you like a step-by-step explanation of the solution process? ### Related Questions: 1. How do you solve a system of equations using the substitution method? 2. What is the determinant of the coefficient matrix, and what does it indicate? 3. Can this system be solved using Cramer's Rule? 4. How do you solve a system of equations using row reduction (Gaussian elimination)? 5. What happens if a system of equations has no solution or infinitely many solutions? #### Tip: For larger systems, solving with matrices (Gaussian elimination or inverse matrices) is often more efficient than substitution or elimination methods.

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Linear Algebra
Systems of Equations
Matrix Methods

Ax = B (Matrix Form)
Gaussian Elimination
Determinant and Cramer's Rule

Cramer's Rule
Properties of Determinants
Inverse Matrix Theorem

Grades 10-12, College Algebra

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