Question

If the system of linear equations $x + 2y + z = 5, 2x + \lambda y + 4z = 12, 4x + 8y + 12z = 2\mu$ have infinite number of solutions, then the values of $\lambda$ and $\mu$ are ________?
 

• A. $\lambda = 4, \mu = 28$
• B. $\lambda = 4, \mu = 14$
• C. $\lambda = 8, \mu = 14$
• D. $\lambda = 8, \mu = 28$

Hint:

For systems of linear equations with infinite solutions, check the determinant of the coefficient matrix for consistency.

Answer 1

The Correct Option is B

For the system to have infinite solutions, the determinant of the coefficient matrix must be zero. The augmented matrix form is: \[ \begin{bmatrix} 1 & 2 & 1 & | & 5 \\ 2 & \lambda & 4 & | & 12 \\ 4 & 8 & 12 & | & 2\mu \end{bmatrix} \] By solving this system and applying the condition for infinite solutions, we get \(\lambda = 4\) and \(\mu = 14\).