Math Problem Statement
Use the Gaussian elimination method to solve the system of linear equations. If the system has infinitely many solutions, write the solution set with z arbitrary. 4 x plus 11 y minus 20 z equals 1 2 x plus 7 y minus 12 z equals negative 3 x plus 2 y minus 4 z equals 2 Question content area bottom Part 1 Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution is StartSet left parenthesis nothing comma nothing comma nothing right parenthesis EndSet . (Simplify your answers. Use integers or fractions for any numbers in the expressions.) B. There are infinitely many solutions. The solution is StartSet left parenthesis nothing comma nothing comma z right parenthesis EndSet , where z is any real number. (Type expressions using z as the variable. Simplify your answers. Use integers or fractions for any numbers in the expressions.) C. The system is inconsistent. The solution set is empty set.
Solution
To solve the system of linear equations using Gaussian elimination, we represent the system as an augmented matrix, perform row reduction to echelon form, and analyze the result.
Step 1: Write the system as an augmented matrix
The system of equations is:
The augmented matrix is:
4 & 11 & -20 & 1 \\ 2 & 7 & -12 & -3 \\ 1 & 2 & -4 & 2 \end{bmatrix}$$ --- ### Step 2: Perform Gaussian elimination #### First, make the pivot in the first column a 1 by dividing the third row by 1: $$\begin{bmatrix} 4 & 11 & -20 & 1 \\ 2 & 7 & -12 & -3 \\ 1 & 2 & -4 & 2 \end{bmatrix}$$ #### Reduce numpy as np from sympy import symbols, Eq, solve, Matrix # Define the augmented matrix matrix = Matrix([ [4, 11, -20, 1], [2, 7, -12, -3], [1, 2, -4, 2] ]) # Perform Gaussian elimination reduced_matrix = matrix.rref() reduced_matrixAfter performing Gaussian elimination, the augmented matrix in row-reduced echelon form is: $$\begin{bmatrix} 1 & 0 & -\frac{4}{3} & \frac{20}{3} \\ 0 & 1 & -\frac{4}{3} & -\frac{7}{3} \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ --- ### Step 3: Interpret the result - The third row, $$[0, 0, 0, 0]$$, indicates that the system is consistent, and there are infinitely many solutions. - Variables $$x$$ and $$y$$ can be expressed in terms of $$z$$, which is arbitrary. --- ### Step 4: Express the solution From the row-reduced form: 1. From the first row: $$x - \frac{4}{3}z = \frac{20}{3} \implies x = \frac{20}{3} + \frac{4}{3}z$$ 2. From the second row: $$y - \frac{4}{3}z = -\frac{7}{3} \implies y = -\frac{7}{3} + \frac{4}{3}z$$ Let $$z = z$$ (arbitrary), then: $$\text{Solution: } \{(x, y, z) \mid x = \frac{20}{3} + \frac{4}{3}z, \, y = -\frac{7}{3} + \frac{4}{3}z, \, z \in \mathbb{R}\}.$$ --- ### Final Answer: **Choice B:** There are infinitely many solutions. The solution is: $$\{(x, y, z) \mid x = \frac{20}{3} + \frac{4}{3}z, \, y = -\frac{7}{3} + \frac{4}{3}z, \, z \in \mathbb{R}\}.$$ Let me know if you'd like further clarifications or steps! #### Related Questions: 1. How do we determine if a system has no solution or infinitely many solutions using Gaussian elimination? 2. What is the importance of the row-reduced echelon form in solving systems of equations? 3. Can you provide a geometrical interpretation of this system’s solution set? 4. How would the solution change if one equation was modified slightly? 5. What are some applications of Gaussian elimination in real-world problems? #### Tip: When solving systems of linear equations, always check for consistency by analyzing the rows of the reduced matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Linear Equations
Formulas
Row-reduction method for solving Ax = b
General representation of solutions for consistent systems
Theorems
Gaussian Elimination Method
Consistency and Solution Classification of Linear Systems
Suitable Grade Level
Grades 10-12, Undergraduate
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