Math Problem Statement

Solve the system of linear equations with parameters m and k. The system consists of four linear equations with four variables.

Solution

The problem provided involves solving a system of linear equations, and consists of three parts. Let's first break down the system of equations:

&x_1 + 2x_2 - x_3 + mx_4 = 4 \\ &-x_1 - x_2 + 3x_3 + 2x_4 = k \\ &2x_1 - x_2 - 3x_3 + (m - 1)x_4 = 3 \\ &x_1 + x_2 + x_3 + 2mx_4 = 5 \end{aligned}$$ ### Part a) Solve the system when $$m = 2$$ and $$k = 5$$. Substitute $$m = 2$$ and $$k = 5$$ into the system: $$\begin{aligned} &x_1 + 2x_2 - x_3 + 2x_4 = 4 \\ &-x_1 - x_2 + 3x_3 + 2x_4 = 5 \\ &2x_1 - x_2 - 3x_3 + x_4 = 3 \\ &x_1 + x_2 + x_3 + 4x_4 = 5 \end{aligned}$$ This is now a system of four linear equations with four variables. Solving this system involves either Gaussian elimination or matrix methods. ### Part b) Conditions for the system to have a unique solution. For the system to have a unique solution, the determinant of the coefficient matrix must be non-zero. This can be found by analyzing the matrix: $$\begin{bmatrix} 1 & 2 & -1 & m \\ -1 & -1 & 3 & 2 \\ 2 & -1 & -3 & (m-1) \\ 1 & 1 & 1 & 2m \end{bmatrix}$$ Set the determinant of this matrix not equal to zero to find the values of $$m$$ and $$k$$ where the system has a unique solution. ### Part c) Conditions for the system to have infinitely many solutions. To have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the system must be consistent. This involves solving the determinant of the matrix and analyzing the conditions for which the system becomes dependent (e.g., one or more rows become linearly dependent). --- Would you like a detailed solution for part (a) using matrix methods or Gaussian elimination? Additionally, here are a few related questions for deeper exploration: 1. How do we compute the determinant of a 4x4 matrix efficiently? 2. What are the conditions for a system of equations to be inconsistent? 3. What methods are best for solving large systems of linear equations? 4. How do we interpret the geometric meaning of a unique solution vs. infinite solutions in linear systems? 5. Can a system have no solution even if the determinant is zero? **Tip:** When solving systems of equations, always check the determinant of the coefficient matrix first to determine if a unique solution exists.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Theory

Formulas

Determinant of a matrix
Gaussian elimination

Theorems

Unique solution condition (determinant ≠ 0)
Infinite solutions condition (determinant = 0, consistent system)

Suitable Grade Level

Undergraduate level, Linear Algebra

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