If the system of linear equations x - 8y + 7z = g, 3y - 9z = h, and -6x + 5y - z = k is consistent, then:

The given system of linear equations is:
x - 8y + 7z = g
3y - 9z = h
-6x + 5y - z = k
For the system to be consistent, the determinant of the coefficient matrix must be zero.
The coefficient matrix is:
| 1 -8 7 |
| 0 3 -9 |
| -6 5 -1 |
The determinant is:
1(3(-1) - (-9)(5)) - (-8)(0(-1) - (-9)(-6)) + 7(0(5) - 3(-6))
= 1(-3 + 45) + 8(0 - 54) + 7(0 + 18)
= 42 - 432 + 126
= -264 ≠ 0
Since the determinant is not zero, the system of equations is inconsistent unless g, h, and k satisfy a specific condition.
Let's use elimination to find the relationship between g, h, and k.
From the second equation, 3y = h + 9z, so y = (h + 9z)/3.
Substitute this into the first equation:
x - 8((h + 9z)/3) + 7z = g
x - (8h/3) - 24z + 7z = g
x = g + (8h/3) + 17z
Substitute into the third equation:
-6(g + (8h/3) + 17z) + 5((h + 9z)/3) - z = k
-6g - 16h - 102z + (5h/3) + 15z - z = k
-6g - 16h - 102z + (5h/3) + 14z = k
-6g - (43h/3) - 88z = k
This equation shows the relationship between g, h, k, and z.
However, there must be a mistake in the problem statement or the provided options. The determinant of the coefficient matrix being non-zero implies that there's a unique solution only if the system is consistent. The options suggest a linear relationship between g, h, and k which is not possible without further constraint. There is no solution among the given options. Let's re-examine the equations and ensure no error in transcription.