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 if the system is homogeneous, or the augmented matrix must satisfy specific conditions. Here, we can perform row operations to eliminate variables and find the condition on g, h, and k.
The system is: <br> x - 4y + 7z = g (1) <br> 3y - 5z = h (2) <br> -2x + 5y - 9z = k (3) <br> Adding 2 times equation (1) to equation (3): <br> 2(x - 4y + 7z) + (-2x + 5y - 9z) = 2g + k <br> -3y + 5z = 2g + k <br> From equation (2), we have 3y - 5z = h, which means -3y + 5z = -h. <br> Therefore, -h = 2g + k, which implies 2g + h + k = 0.
Correct Answer: (2)