Question

If the system of equations \[ x + y + z = 5 \] \[ x + 2y + 3z = 9 \] \[ x + 3y + \lambda z = \mu \] has infinitely many solutions, then value of \( \lambda + \mu \) is

• A. \(13\)
• B. \(20\)
• C. \(18\)
• D. \(26\)

Hint:

A system of three linear equations has infinitely many solutions when: \[ D = D_1 = D_2 = D_3 = 0 \] which means the equations are dependent.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix (\( \Delta \)) must be zero, and the specific determinants \( \Delta_x, \Delta_y, \Delta_z \) must also be zero. This happens when the third equation is a linear combination of the first two.
Step 2: Key Formula or Approach:
1. Set the determinant of coefficients to zero to find \( \lambda \). 2. Use the augmented matrix or the condition that the rows are linearly dependent to find \( \mu \).
Step 3: Detailed Explanation:
1. Check the pattern of the equations: Eq 1: \( x + y + z = 5 \) Eq 2: \( x + 2y + 3z = 9 \) Notice that Eq 2 - Eq 1 gives: \( y + 2z = 4 \). If we add this difference again to Eq 2: \( (x + 2y + 3z) + (y + 2z) = 9 + 4 \) \( x + 3y + 5z = 13 \). 2. Compare this to the third given equation \( x + 3y + \lambda z = \mu \): By comparison, \( \lambda = 5 \) and \( \mu = 13 \). 3. Find the sum: \( \lambda + \mu = 5 + 13 = 18 \).
Step 4: Final Answer:
The value of \( \lambda + \mu \) is 18.