Final answer:
The system of linear equations is inconsistent if there is a contradiction. In this case, the system is inconsistent for the value of k = 9 because that value leads to a statement of 0 = 23 which is a contradiction.
Step-by-step explanation:
To determine for what value(s) of k the given system of linear equations (SOLE) is inconsistent, we need to set up the system's augmented matrix and perform Gaussian elimination. The system is inconsistent if and only if there are any rows of the form [0 0 0 | 1], as this would imply a contradiction (0 = 1).
The augmented matrix for the SOLE is:
[-2 3 7 | 4]
[6 k -5 |-11]
[0 5 2 | 7]
Now, perform Gaussian elimination.
- Multiply the first row by 3 and the second row by -1, then add the two rows together. The system becomes:
[6 9 21 | 12]
[-6 -k 5 | 11]
[0 5 2 | 7] - Add the first and second rows together: [0 9-k 26 | 23]
[-6 -k 5 | 11]
[0 5 2 | 7]
If 9 - k equals 0, the system is inconsistent, because we will have a row of [0 0 26 | 23], implying that 0 = 23, a contradiction.
So, the system is inconsistent for k = 9.
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