Math Problem Statement

Test for what values of λ and μ the system of equations x + y + z = 6, x + 2y + 3z = 10, and x + 2y + λz = μ has no solution, a unique solution, or infinite solutions.

Solution

This is a question regarding the system of linear equations:

  1. x+y+z=6x + y + z = 6
  2. x+2y+3z=10x + 2y + 3z = 10
  3. x+2y+λz=μx + 2y + \lambda z = \mu

We are tasked to determine the values of λ\lambda and μ\mu for which the system of equations has:

  1. No solution
  2. A unique solution
  3. Infinite solutions

Step-by-step solution:

1. Analyze the system as a matrix:

The augmented matrix for the system is:

1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 10 \\ 1 & 2 & \lambda & \mu \end{bmatrix}$$ #### 2. Perform row operations to identify conditions: - Subtract the first row from the second and third rows to simplify: $$\begin{bmatrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 1 & \lambda - 1 & \mu - 6 \end{bmatrix}$$ - Subtract the second row from the third row: $$\begin{bmatrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & (\lambda - 1) - 2 & (\mu - 6) - 4 \end{bmatrix}$$ Simplify the third row: $$\begin{bmatrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & \lambda - 3 & \mu - 10 \end{bmatrix}$$ #### 3. Determine the conditions: - **Unique solution**: For a unique solution, the determinant of the coefficient matrix must be non-zero. This occurs when $$\lambda - 3 \neq 0$$. Thus, $$\lambda \neq 3$$. - **Infinite solutions**: For infinite solutions, the third row must become a row of zeros. This happens when: $$\lambda - 3 = 0 \quad \text{and} \quad \mu - 10 = 0$$ Hence, $$\lambda = 3$$ and $$\mu = 10$$. - **No solution**: For no solution, the third row must result in a contradiction. This happens when: $$\lambda - 3 = 0 \quad \text{and} \quad \mu - 10 \neq 0$$ Hence, $$\lambda = 3$$ and $$\mu \neq 10$$. --- ### Final Answer: 1. **No solution**: $$\lambda = 3$$ and $$\mu \neq 10$$ 2. **Unique solution**: $$\lambda \neq 3$$ 3. **Infinite solutions**: $$\lambda = 3$$ and $$\mu = 10$$ Let me know if you need further clarifications or a detailed explanation! --- **5 Related Questions**: 1. How can you determine the rank of a matrix for a system of linear equations? 2. What are the geometric interpretations of no solution, unique solution, and infinite solutions? 3. How does the determinant of a matrix relate to the uniqueness of solutions in a system? 4. Can you solve a similar problem using row echelon form? 5. What happens if the augmented matrix's rank differs from the coefficient matrix's rank? **Tip**: Always check the determinant of the coefficient matrix first to determine if a system is solvable or has unique solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrices
Row Reduction

Formulas

Row operations for solving linear systems
Condition for unique solutions: Determinant of the coefficient matrix ≠ 0
Condition for infinite solutions: Rank of coefficient matrix = Rank of augmented matrix
Condition for no solution: Rank of coefficient matrix < Rank of augmented matrix

Theorems

Rouché–Capelli Theorem

Suitable Grade Level

Grades 11-12 or Undergraduate

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