Final answer:
The system will be consistent for any value of k.
Step-by-step explanation:
The given system of equations is:
6x + 4y - 5z = 7
7x + 2y + kz = -5
32x + 16y - z = 6
To check if the system is consistent, we can eliminate variables x and y by multiplying the first equation by 2 and the second equation by 3. This will give us:
12x + 8y - 10z = 14
21x + 6y + 3kz = -15
Subtracting the first equation from the second equation:
9x - 14z = -29
From the third equation, we can write:
32x + 16y = z + 6
Substituting the value of z from the second equation into the first equation:
9x - 14(32x + 16y - 6) = -29
Simplifying:
9x - 448x - 224y + 84 = -29
-439x - 224y = -113
Now we have a system of two equations:
9x - 14z = -29
-439x - 224y = -113
If the system is consistent, there will be a unique solution for x and y. To check if the system has a unique solution, we can calculate the determinant:
D = (9)(-224) - (-14)(-439)
D = 1996
Since the determinant is not zero, the system has a unique solution for x and y. Therefore, for the system to be consistent, k can be any value.