Question

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

• A. 16
• B. 18
• C. 19
• D. 21

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a system to have infinitely many solutions, the determinant of the coefficient matrix (\(\Delta\)) must be zero, and the constants must be consistent with the linear combination of the other equations.

Step 2: Key Formula or Approach:
1. Set \(\Delta = 0\).
2. Use row transformations to check consistency or set \(\Delta_x = \Delta_y = \Delta_z = 0\).

Step 3: Detailed Explanation:
\(\Delta = \begin{vmatrix} 1 & 1 & 1 \\1 & 2 & 3 \\ 1 & 3 & \lambda \end{vmatrix} = 0 \).
\(R_2 \to R_2 - R_1, R_3 \to R_3 - R_2 \):
\(\begin{vmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 1 & \lambda - 3 \end{vmatrix} = 0 \implies 1(\lambda - 3 - 2) = 0 \implies \lambda = 5\).
For infinite solutions, the constant terms must follow the same row relationship.
Performing row operations on the augmented matrix \([A|B]\):
\(R_2 \to R_2 - R_1 \implies 0x + 1y + 2z = 4\).
\(R_3 \to R_3 - R_2 \implies 0x + 1y + (\lambda - 3)z = \mu - 9\).
For these to be identical: \(\lambda - 3 = 2 \implies \lambda = 5\) and \(\mu - 9 = 4 \implies \mu = 13\).
\(\lambda + \mu = 5 + 13 = 18\).

Step 4: Final Answer:
The value is 18.