Question

Consider the system of equations
\(x + y + z = 6\)
\(x + 2y + 5z = 18\)
\(2x + 2y + \lambda z = \mu\)
If the system of equations has infinitely many solutions, then the value of \((\lambda + \mu)\) is equal to

Answer 1

Correct Answer: 14

Step 1: Understanding the Concept:
For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix ($Δ$) must be zero, and all specific determinants ($\Deltaₓ, \Deltay, \Deltaz$) must also be zero. Alternatively, for three equations in three variables, the third equation must be a linear combination of the first two.

Step 2: Key Formula or Approach:
We compare the third equation with the first two. Notice that the coefficients of $x$ and $y$ in the third equation ($2x + 2y$) are exactly double the coefficients of $x$ and $y$ in the first equation ($x + y$). For the system to be consistent with infinitely many solutions, the third equation must be a direct multiple of the first equation, or the second equation must not contradict this relationship.

Step 3: Detailed Explanation:
1. Let the equations be: (i) $x + y + z = 6$ (ii) $x + 2y + 5z = 18$ (iii) $2x + 2y + λ z = μ$ 2. For infinitely many solutions, $Δ = 0$: \[ \begin{vmatrix} 1 & 1 & 1 1 & 2 & 5 2 & 2 & \lambda \end{vmatrix} = 0 \] 3. Expanding the determinant: \[ 1(2\lambda - 10) - 1(\lambda - 10) + 2(5 - 2) = 0 \] \[ 2\lambda - 10 - \lambda + 10 + 6 = 0 \] \[ \lambda + 6 = 0 \implies \lambda = 2 \] *(Note: Multiplying the first row by 2 makes the first two elements identical to the third row. Thus, for the determinant to be zero, $λ$ must be $1 × 2 = 2$).* 4. For $\Deltaₓ, \Deltay, \Deltaz$ to be zero, the constant terms must follow the same ratio. Since the third row is twice the first row in its $x$ and $y$ coefficients, the constant $μ$ must be twice the constant of the first equation: \[ \mu = 2 \times 6 = 12 \] 5. Check consistency with Eq (ii): Since Eq (iii) is now just $2x + 2y + 2z = 12$, it is equivalent to Eq (i). We are left with two independent equations, which always yield infinitely many solutions in three variables. 6. Calculate $λ + μ$: \[ 2 + 12 = 14 \]

Step 4: Final Answer:
The value of $(λ + μ)$ is 14.