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

We are given the system of equations: \[ x + y + z = 6, \] \[ x + 2y + 5z = 9, \] \[ x + 5y + \lambda z = \mu. \] - We can solve this system by using elimination or substitution to obtain the conditions under which the system has no solution. For a system to have no solution, the determinant of the coefficient matrix must be zero, or the equations must be inconsistent. - After solving the system, we find that the system will have no solution when \( \lambda = 17 \) and \( \mu \neq 18 \). Conclusion: The correct answer is (1), as the system has no solution when \( \lambda = 17 \) and \( \mu \neq 18 \).