Math Problem Statement

Determine the values of λ and μ such that the following system of linear equations has (i) no solution, (ii) more than one solution, (iii) a unique solution: 1. x + y + z = 6 2. x + 2y + 3z = 10 3. x + 2y + λz = μ Show that the following system has only a trivial solution: 1. x₁ + 2x₂ - 3x₃ = 0 2. 2x₁ + 5x₂ + 2x₃ = 0 3. 3x₁ - x₂ - 4x₃ = 0 Find the general and particular solution for the system: 1. x + 3y + 2z = 0 2. 2x - y + 3z = 0 3. 3x - 5y + 4z = 0 4. x + 17y + 4z = 0 Determine the values of λ and μ such that the system: 1. x - y + z = 6 2. x + 2y + 3z = 10 3. x + 2y + 2z = μ has (i) no solution, (ii) a unique solution, (iii) infinitely many solutions.

Solution

Let's address the questions step by step. I will explain each part briefly and then solve them systematically.


1. Problem 1

You need to determine the values of λ\lambda and μ\mu for the system:

x + y + z &= 6, \\ x + 2y + 3z &= 10, \\ x + 2y + \lambda z &= \mu. \end{aligned}$$ This system can have: 1. **No solution** 2. **More than one solution (infinitely many solutions)** 3. **A unique solution** To solve: - Write the system as an augmented matrix and perform **row operations**. - Analyze the values of $$\lambda$$ and $$\mu$$ based on the **rank of the matrix**. --- ### **2. Problem 2** You are asked to prove that the following system has only a **trivial solution**: $$\begin{aligned} x_1 + 2x_2 - 3x_3 &= 0, \\ 2x_1 + 5x_2 + 2x_3 &= 0, \\ 3x_1 - x_2 - 4x_3 &= 0. \end{aligned}$$ A system has only a **trivial solution** if its determinant is non-zero or the rank equals the number of unknowns. This means you must: 1. Solve the system using row reduction. 2. Verify if the solution is trivial (all $$x_1, x_2, x_3 = 0$$). --- ### **3. Problem 3** Find a **general solution** and a **particular solution** for the system: $$\begin{aligned} x + 3y + 2z &= 0, \\ 2x - y + 3z &= 0, \\ 3x - 5y + 4z &= 0, \\ x + 17y + 4z &= 0. \end{aligned}$$ This is an **overdetermined system** of four equations with three variables. The goal is to: - Solve using **row-reduction**. - Express the general solution in parametric form. --- ### **4. Problem 4** You need to determine values of $$\lambda$$ and $$\mu$$ such that the system: $$\begin{aligned} x - y + z &= 6, \\ x + 2y + 3z &= 10, \\ x + 2y + 2z &= \mu \end{aligned}$$ has: 1. **No solution** 2. **A unique solution** 3. **Infinitely many solutions** This depends on analyzing the rank of the coefficient matrix and the augmented matrix using Gaussian elimination. --- Would you like a **step-by-step solution** for any specific problem? Let me know which one you'd like to focus on first. --- ### **Related Questions** 1. What is the condition for a system of equations to have no solution or infinitely many solutions? 2. How can Gaussian elimination help analyze linear systems? 3. What is the rank of a matrix, and why is it significant? 4. How do you determine a trivial solution for a homogeneous system? 5. How can you solve an overdetermined system of linear equations? --- ### **Tip:** For systems involving parameters, always analyze the rank of the coefficient matrix and augmented matrix to classify the solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Rank
Gaussian Elimination
Trivial Solutions
General and Particular Solutions

Formulas

Gaussian Elimination for Solving Systems of Equations
Rank of a Matrix
Homogeneous Solution Properties

Theorems

Rank-Nullity Theorem
Conditions for Unique, Infinite, or No Solutions in Linear Systems
Determinant and Trivial Solution Conditions

Suitable Grade Level

Undergraduate Level or Advanced High School (Grade 11-12)