Question

The system of equations \[ x + y + z = 6, \] \[ x + 2y + 5z = 9, \] \[ x + 5y + \lambda z = \mu, \] has no solution if:

• A. \( \lambda = 17, \mu \neq 18 \)
• B. \( \lambda \neq 17, \mu \neq 18 \)
• C. \( \lambda = 15, \mu \neq 17 \)
• D. \( \lambda = 17, \mu = 18 \)

Hint:

For systems of linear equations, use substitution or elimination to simplify and solve. Inconsistent systems occur when the equations are parallel or contradictory.

Answer 1

The Correct Option is A

Given the system of equations: \[ x + y + z = 6, \] \[ x + 2y + 5z = 9, \] \[ x + 5y + \lambda z = \mu. \]
- The system can be solved using elimination or substitution to determine the conditions for no solution. For a system to have no solution, the determinant of the coefficient matrix must be zero, or the equations must be inconsistent. 
- Upon solving, it is determined that the system has no solution when \( \lambda = 17 \) and \( \mu eq 18 \). 

Conclusion: The system has no solution when \( \lambda = 17 \) and \( \mu eq 18 \), thus option (1) is correct.