If the system of linear equations <br> x - 4y + 7z = g <br> 3y - 5z = h <br> -2x + 5y - 9z = k <br> is consistent, then :
Step-by-Step Solution
Key Concept: For a system of linear equations to be consistent, the determinant of the coefficient matrix must be zero, and the augmented matrix must satisfy the consistency condition (rank(A) = rank(A|B)).
The system is consistent if the determinant of the coefficient matrix is zero and the augmented matrix satisfies the condition. The coefficient matrix is A = [[1, -4, 7], [0, 3, -5], [-2, 5, -9]]. The determinant |A| = 1(-27 + 25) - (-4)(0 - 10) + 7(0 + 6) = -2 - 40 + 42 = 0. Since |A| = 0, the system is consistent if the augmented matrix [A|B] has the same rank as A. Performing row operations on the augmented matrix [1, -4, 7, g; 0, 3, -5, h; -2, 5, -9, k] -> R3 -> R3 + 2R1 gives [1, -4, 7, g; 0, 3, -5, h; 0, -3, 5, k + 2g]. Adding R2 to R3 gives [0, 0, 0, k + 2g + h]. For consistency, k + 2g + h = 0.
Correct Answer: 2