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

Concept: For a system of linear equations to have infinitely many solutions: \[ D = D_1 = D_2 = D_3 = 0 \] where \(D\) is the determinant of the coefficient matrix and \(D_1, D_2, D_3\) are obtained by replacing respective columns with constants.
Step 1: Coefficient determinant \[ D = \begin{vmatrix} 1 & 1 & 1
1 & 2 & 3
1 & 3 & \lambda \end{vmatrix} \] \[ D = 1(2\lambda - 9) - 1(\lambda - 3) + 1(3 - 2) \] \[ D = 2\lambda - 9 - \lambda + 3 + 1 \] \[ D = \lambda - 5 \] For infinitely many solutions, \[ D = 0 \Rightarrow \lambda = 5 \]
Step 2: Calculate \(D_1\) \[ D_1 = \begin{vmatrix} 5 & 1 & 1
9 & 2 & 3
\mu & 3 & \lambda \end{vmatrix} \] Substituting \( \lambda = 5 \): \[ D_1 = \lambda + \mu - 18 \] For infinite solutions: \[ D_1 = 0 \] \[ \lambda + \mu = 18 \]
Step 3: Other determinants \[ D_2 = 4\lambda - 2\mu + 6 \] \[ D_3 = \mu - 13 \] These also become zero when \[ \lambda = 5, \quad \mu = 13 \]
Step 4: Final value \[ \lambda + \mu = 5 + 13 = 18 \]