Math Problem Statement

Solve the following systems of linear equations: a) 3x₁ - 5x₂ + 2x₃ + 4x₄ = 2 7x₁ - 4x₂ + x₃ + 3x₄ = 5 5x₁ + 7x₂ - 4x₃ - 6x₄ = 3 b) 3x₁ - x₂ + 3x₃ = 1 -4x₁ + 2x₂ + x₃ = 3 -2x₁ + x₂ + 4x₃ = 4 10x₁ - 5x₂ - 6x₃ = -10

Solution

Let's solve these two systems of linear equations.

Part (a):

The given system of equations is:

3x_1 - 5x_2 + 2x_3 + 4x_4 = 2 \\ 7x_1 - 4x_2 + x_3 + 3x_4 = 5 \\ 5x_1 + 7x_2 - 4x_3 - 6x_4 = 3 \\ \end{cases}$$ ### Part (b): The given system of equations is: $$\begin{cases} 3x_1 - x_2 + 3x_3 = 1 \\ -4x_1 + 2x_2 + x_3 = 3 \\ -2x_1 + x_2 + 4x_3 = 4 \\ 10x_1 - 5x_2 - 6x_3 = -10 \\ \end{cases}$$ I'll solve these systems using Gaussian elimination. Let me calculate the solutions step by step.For system (a), it appears that the matrix is not square (3 equations for 4 variables), meaning a unique solution may not exist, and additional methods or constraints would be required to find a solution. For system (b), using the least squares method yields the solution: $$x_1 \approx 0, \quad x_2 \approx 1.143, \quad x_3 \approx 0.714$$ These values provide an approximate solution that minimizes the error for the overdetermined system. Would you like more details, or do you have any questions about these solutions? ### Related Questions: 1. How does Gaussian elimination work for solving linear systems? 2. What are the conditions for a unique solution in linear equations? 3. How does the least squares method solve overdetermined systems? 4. What alternative methods can be used for solving non-square systems? 5. How does matrix rank affect the solvability of linear systems? ### Tip: When dealing with systems where the number of equations does not match the number of variables, consider using least squares or finding the null space to understand the solution set better.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Overdetermined Systems

Formulas

Gaussian Elimination steps
Least Squares method for overdetermined systems

Theorems

Gaussian Elimination
Least Squares Approximation

Suitable Grade Level

Undergraduate

Related Recommendation

Solve System of Linear Equations with Gaussian Elimination

Solve the System of Linear Equations Using Gaussian Elimination

Solve a System of Linear Equations Using Gaussian Elimination

Solving a System of Linear Equations Using Gaussian Elimination

Solve Inconsistent Linear Systems Using Least-Squares Approximation