Main Answer:
The system's consistency relies on the determinant; k ≠9 ensures a nonzero determinant, securing a unique solution's existence. D. 9.
Therefore, the correct answer is D. 9.
Step-by-step explanation:
For a system of linear equations to be consistent, it must have a unique solution or an infinite number of solutions. The key lies in the determinant of the coefficient matrix. In this case, the determinant is obtained by setting up the matrix formed by the coefficients of the variables and evaluating it. If the determinant is nonzero, the system is consistent. Setting up and solving the determinant for the given system, we find that for the system to be consistent, k must not be equal to 9.
In the third equation, if k is 9, the determinant becomes zero, indicating that the system may have no solution or infinitely many solutions, rendering it inconsistent. Therefore, to maintain consistency, the value of k must not be 9.
Consistency in a system of linear equations is crucial for meaningful solutions. When the determinant is nonzero, it signifies that the system has a unique solution. A determinant of zero suggests dependency among the equations, leading to an inconsistent or dependent system. Therefore, to ensure the given system is consistent, the value of k must not be 9.
Therefore, the correct answer is D. 9.