For the system of linear equation<br>2x - y + 3z = 5<br>3x + 2y - z = 7<br>4x + 5y + αz = β<br>Which of the following is(are) CORRECT?
(A) The system has infinitely many solutions for α = -5 and β = 9
(B) The system has a unique solution for α ≠ -5 and β = 8
(C) The system has infinitely many solutions for α = -6 and β = 9
(D) The system is inconsistent for α = -5 and β = 8
Step-by-Step Solution
Key Concept: Calculate the determinant of the coefficient matrix D. If D = 0, check for consistency using Cramer's rule or augmented matrix rank. If D ≠ 0, the system has a unique solution.
The coefficient matrix is A = [[2, -1, 3], [3, 2, -1], [4, 5, α]]. The determinant D = 2(2α + 5) + 1(3α + 4) + 3(15 - 8) = 4α + 10 + 3α + 4 + 21 = 7α + 35. D = 0 when α = -5. For α = -5, the system is 2x - y + 3z = 5, 3x + 2y - z = 7, 4x + 5y - 5z = β. Adding the first two equations multiplied by constants to eliminate variables shows that for β = 9, the system is consistent with infinite solutions. For β = 8, the system is inconsistent. For α ≠ -5, D ≠ 0, so the system has a unique solution.
Correct Answer: 1, 2, 4