Math Problem Statement

Solve the system of linear equations provided in the image: x1 + x2 + x3 - x4 = 1, 3x1 + x2 + 2x3 - 2x4 = 2, 2x1 - 4x2 - 3x3 + 6x4 = 7, 7x1 + 5x2 + 6x3 - 6x4 = 6.

Solution

The system of linear equations given in the image is as follows:

1. & \quad x_1 + x_2 + x_3 - x_4 = 1, \\ 2. & \quad 3x_1 + x_2 + 2x_3 - 2x_4 = 2, \\ 3. & \quad 2x_1 - 4x_2 - 3x_3 + 6x_4 = 7, \\ 4. & \quad 7x_1 + 5x_2 + 6x_3 - 6x_4 = 6. \end{aligned}$$ ### Solution Steps 1. Represent this system in augmented matrix form. 2. Use Gaussian elimination or matrix methods (such as row reduction or inverse matrices) to solve. 3. Provide the solution for $$x_1, x_2, x_3, x_4$$. Let me calculate the solution. numpy as np # Coefficients of the equations A = np.array([ [1, 1, 1, -1], [3, 1, 2, -2], [2, -4, -3, 6], [7, 5, 6, -6] ]) # Constants on the right-hand side B = np.array([1, 2, 7, 6]) # Solve the system of equations solution = np.linalg.solve(A, B) solutionThe matrix of the given system is singular, which means it does not have a unique solution. This typically occurs when the equations are either dependent or inconsistent. To further analyze, I will determine whether the system is consistent and find all possible solutions if they exist using reduced row echelon form (RREF). Let me proceed. sympy import Matrix # Convert the system into a sympy Matrix for RREF computation augmented_matrix = Matrix([ [1, 1, 1, -1, 1], [3, 1, 2, -2, 2], [2, -4, -3, 6, 7], [7, 5, 6, -6, 6] ]) # Compute the Reduced Row Echelon Form (RREF) rref_matrix, pivot_columns = augmented_matrix.rref() rref_matrixThe reduced row echelon form (RREF) of the augmented matrix is: $$\begin{bmatrix} 1 & 0 & 0 & \frac{3}{4} & \frac{5}{2} \\ 0 & 1 & 0 & \frac{3}{4} & \frac{5}{2} \\ 0 & 0 & 1 & -\frac{5}{2} & -4 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}$$ ### Interpretation: - The system has a free variable ($$x_4$$), as the last column of the augmented matrix does not lead to any inconsistency. - This means there are infinitely many solutions, and $$x_4$$ can take any value. ### General Solution: $$\begin{aligned} x_1 &= \frac{5}{2} - \frac{3}{4}x_4, \\ x_2 &= \frac{5}{2} - \frac{3}{4}x_4, \\ x_3 &= -4 + \frac{5}{2}x_4, \\ x_4 &= x_4 \, (\text{free variable}). \end{aligned}$$ Would you like further explanation on how to interpret the general solution, or explore specific values for $$x_4$$? Let me know! --- Here are 5 questions to expand understanding: 1. What is the significance of a free variable in a system of linear equations? 2. How can the solution be interpreted geometrically in four dimensions? 3. How would this solution change if an additional constraint were applied? 4. Can the solution be represented parametrically? 5. What does it mean when the coefficient matrix is singular? **Tip:** In systems with free variables, expressing solutions parametrically can help visualize relationships between variables.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Gaussian Elimination
Matrix Theory

Formulas

Ax = B for solving systems of equations
Gaussian Elimination for row reduction
Reduced Row Echelon Form (RREF)

Theorems

Rank-Nullity Theorem
Existence of solutions based on matrix consistency

Suitable Grade Level

College Level or Advanced High School (Grades 11-12)