Math Problem Statement
Solve it
Investigate the values of λ and μ so that the equations 2x + 3y + 5z = 9, 7x + 3y - 2z = 8, 2x + 3y + λz = μ have (a) no solution, (b) a unique solution, and (c) an infinite number of solutions.
Solution
The problem asks to investigate the values of and such that the following system of equations:
2x + 3y + 5z &= 9 \\ 7x + 3y - 2z &= 8 \\ 2x + 3y + \lambda z &= \mu \end{aligned}$$ has: - (a) no solution, - (b) a unique solution, and - (c) an infinite number of solutions. ### Solution Outline: This system can be studied using methods of linear algebra, particularly by analyzing the consistency of the system. Let’s consider the augmented matrix corresponding to the system: $$\begin{pmatrix} 2 & 3 & 5 & 9 \\ 7 & 3 & -2 & 8 \\ 2 & 3 & \lambda & \mu \end{pmatrix}$$ We will perform row operations on this matrix to determine the conditions for $$\lambda$$ and $$\mu$$ based on whether the system has: 1. No solution (inconsistent system), 2. A unique solution (consistent system with independent equations), 3. An infinite number of solutions (consistent system with dependent equations). ### Step 1: Convert the system to row echelon form We start by eliminating terms below the first pivot element (the first element of the first row). - Subtract row 1 from row 3: $$\begin{aligned} R_3 \rightarrow R_3 - R_1 &= \begin{pmatrix} 2 & 3 & \lambda - 5 & \mu - 9 \end{pmatrix} \end{aligned}$$ We now have the system: $$\begin{aligned} \begin{pmatrix} 2 & 3 & 5 & 9 \\ 7 & 3 & -2 & 8 \\ 0 & 0 & \lambda - 5 & \mu - 9 \end{pmatrix} \end{aligned}$$ ### Step 2: Analyze the system for different conditions - **Case 1: No solution** For the system to have no solution, the third equation should represent an inconsistency (e.g., something like $$0 = k$$ where $$k \neq 0$$). This happens when: $$\lambda - 5 = 0 \quad \text{and} \quad \mu - 9 \neq 0$$ Thus, for no solution: $$\lambda = 5 \quad \text{and} \quad \mu \neq 9$$ - **Case 2: Unique solution** For a unique solution, the system must be consistent, and the coefficient matrix must have full rank, meaning that there should be three independent equations. This happens when the third row provides new information, i.e., $$\lambda - 5 \neq 0$$. Thus, for a unique solution: $$\lambda \neq 5$$ - **Case 3: Infinite number of solutions** For the system to have infinitely many solutions, the third equation must be dependent on the first two, meaning that both: $$\lambda - 5 = 0 \quad \text{and} \quad \mu - 9 = 0$$ Thus, for infinitely many solutions: $$\lambda = 5 \quad \text{and} \quad \mu = 9$$ ### Summary of Conditions: - **No solution**: $$\lambda = 5$$ and $$\mu \neq 9$$ - **Unique solution**: $$\lambda \neq 5$$ - **Infinite solutions**: $$\lambda = 5$$ and $$\mu = 9$$ Let me know if you want more details or further clarification! ### 5 Follow-up Questions: 1. What method can be used to determine the rank of a matrix? 2. How does Gaussian elimination help in solving systems of linear equations? 3. What are the implications of the determinant of a matrix being zero? 4. How would the solution process change if the system had more than three equations? 5. What is the geometric interpretation of a system with infinite solutions? ### Tip: To determine whether a system has unique, infinite, or no solutions, always check the rank of the coefficient matrix compared to the augmented matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrices
Row Echelon Form
Formulas
Augmented matrix
Row reduction
Theorems
Consistency Theorem
Rank Theorem
Suitable Grade Level
University level (Linear Algebra course)
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