Math Problem Statement

For what values of λ and μ does the given system of equations have (a) no solution, (b) a unique solution, and (c) an infinite number of solutions? The equations are: -4x + 2y - 9z = 2, 3x + 4y + z = 5, and 3x + 4y + λz = μ.

Solution

To solve the given problem, we will analyze the system of equations and determine the conditions on λ\lambda and μ\mu for the following cases:

  1. No solution (system is inconsistent).
  2. Unique solution (system is consistent and determinant is non-zero).
  3. Infinite number of solutions (system is consistent with dependent equations).

The system of equations is:

1.\quad & -4x + 2y - 9z = 2, \\ 2.\quad & 3x + 4y + z = 5, \\ 3.\quad & 3x + 4y + \lambda z = \mu. \end{aligned}$$ ### Solution Steps: #### Step 1: Write the system in matrix form The augmented matrix for the system is: $$\begin{bmatrix} -4 & 2 & -9 & 2 \\ 3 & 4 & 1 & 5 \\ 3 & 4 & \lambda & \mu \end{bmatrix}.$$ #### Step 2: Analyze conditions for solutions To analyze, compute the determinant of the coefficient matrix: $$\begin{bmatrix} -4 & 2 & -9 \\ 3 & 4 & 1 \\ 3 & 4 & \lambda \end{bmatrix}.$$ The determinant is given by: $$\text{Det} = -4 \begin{vmatrix} 4 & 1 \\ 4 & \lambda \end{vmatrix} - 2 \begin{vmatrix} 3 & 1 \\ 3 & \lambda \end{vmatrix} - 9 \begin{vmatrix} 3 & 4 \\ 3 & 4 \end{vmatrix}.$$ #### Step 3: Simplify sub-determinants 1. $$\begin{vmatrix} 4 & 1 \\ 4 & \lambda \end{vmatrix} = 4\lambda - 4.$$ 2. $$\begin{vmatrix} 3 & 1 \\ 3 & \lambda \end{vmatrix} = 3\lambda - 3.$$ 3. $$\begin{vmatrix} 3 & 4 \\ 3 & 4 \end{vmatrix} = 0.$$ Substitute back: $$\text{Det} = -4(4\lambda - 4) - 2(3\lambda - 3) - 9(0),$$ $$\text{Det} = -16\lambda + 16 - 6\lambda + 6,$$ $$\text{Det} = -22\lambda + 22.$$ #### Step 4: Conditions on $$\lambda$$ and $$\mu$$ 1. **No solution**: Occurs if $$\text{Det} = 0$$ and the system is inconsistent. $$\text{Det} = 0 \implies -22\lambda + 22 = 0 \implies \lambda = 1.$$ To check inconsistency, substitute $$\lambda = 1$$ into the third equation: $$3x + 4y + z = \mu \quad \text{and} \quad 3x + 4y + z = 5.$$ If $$\mu \neq 5$$, the system is inconsistent. **Condition for no solution**: $$\lambda = 1$$ and $$\mu \neq 5.$$ 2. **Unique solution**: Occurs if $$\text{Det} \neq 0.$$ $$-22\lambda + 22 \neq 0 \implies \lambda \neq 1.$$ **Condition for unique solution**: $$\lambda \neq 1.$$ 3. **Infinite solutions**: Occurs if $$\text{Det} = 0$$ and the third equation is consistent with the first two. $$\text{Det} = 0 \implies \lambda = 1.$$ For consistency, $$\mu = 5.$$ **Condition for infinite solutions**: $$\lambda = 1$$ and $$\mu = 5.$$ ### Final Answer: (a) **No solution**: $$\lambda = 1, \mu \neq 5.$$ (b) **Unique solution**: $$\lambda \neq 1.$$ (c) **Infinite solutions**: $$\lambda = 1, \mu = 5.$$ Would you like a detailed explanation of any step? Let me know! ### Related Questions: 1. What is the geometric interpretation of the determinant in this context? 2. How can we solve the system explicitly for a specific $$\lambda$$ and $$\mu$$? 3. What is the role of the augmented matrix in finding solutions? 4. How does changing $$\lambda$$ affect the solution set? 5. Can you determine if the equations represent planes that are parallel, intersect, or coincide? ### Tip: Always check the determinant of the coefficient matrix first to quickly determine the nature of solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Determinants
Matrix Analysis

Formulas

Determinant of a 3x3 matrix
Conditions for consistency of a system of linear equations (rank conditions)

Theorems

Cramer's Rule
Properties of Determinants

Suitable Grade Level

Grades 11-12 or Undergraduate Level

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