Math Problem Statement
Solution
The task is to solve the given system of linear equations using Gaussian or Gauss-Jordan elimination methods. The system of equations is:
x - 2y + 3z &= 12, \\ 2x + y + z &= 4, \\ -y + 2z &= 7. \end{aligned}$$ ### Step 1: Write the augmented matrix The augmented matrix for this system is: $$\begin{bmatrix} 1 & -2 & 3 & | & 12 \\ 2 & 1 & 1 & | & 4 \\ 0 & -1 & 2 & | & 7 \end{bmatrix}$$ I'll solve this step by step using Gaussian or Gauss-Jordan elimination. sympy as sp # Define the augmented matrix aug_matrix = sp.Matrix([ [1, -2, 3, 12], [2, 1, 1, 4], [0, -1, 2, 7] ]) # Perform Gaussian elimination (row reduction) rref_matrix, pivot_columns = aug_matrix.rref() # Display the result in reduced row-echelon form rref_matrixAfter applying the Gaussian elimination method, the augmented matrix is reduced to: $$\begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & -1 \\ 0 & 0 & 1 & | & 3 \end{bmatrix}$$ ### Solution: From the reduced matrix, we can directly read the solution: $$x = 1, \quad y = -1, \quad z = 3.$$ Would you like a detailed step-by-step explanation of the elimination process? ### 5 Related Questions: 1. How does Gaussian elimination differ from Gauss-Jordan elimination? 2. What are the conditions for a unique solution in a system of linear equations? 3. Can this method be used for systems with no solutions or infinitely many solutions? How? 4. What is the geometric interpretation of solving this system of equations? 5. How can these methods be applied to larger systems of equations? ### Tip: Always check the determinant of the coefficient matrix before solving a system of linear equations—it helps identify if the system is consistent and has a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Gauss-Jordan Elimination
Formulas
Row reduction to echelon form
Backward substitution
Reduced row-echelon form
Theorems
Existence and Uniqueness of Linear Systems Solutions
Suitable Grade Level
Undergraduate Mathematics or Advanced High School
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