For what value of k does the system below not have a unique solution?
-1x + 7y + 2z = 6
2x + 1y + kz = -11
3x - 6y - 2z = -25
A.5
B.0
C.-2
D.-8
determinant = 2+21k-24-6-6k+28
If the determinant is zero, we can not solve for the variables
0 +15 k = 0
k = 0
To determine the value of k for which the given system does not have a unique solution, we need to check if the determinant of the coefficient matrix is equal to zero.
The coefficient matrix (A) of the system is:
A = | -1 7 2 |
| 2 1 k |
| 3 -6 -2 |
To calculate the determinant of A, we can use the formula for a 3x3 matrix:
det(A) = -1(1(-2) - (-6)(k)) - 7(2(-2) - 3(-6)) + 2(2(-6) - 3(1))
det(A) = -(-2 - (-6k)) - 7(-4 + 18) + 2(-12 - 3)
det(A) = 2 + 6k + 7(14) - 2(-15)
det(A) = 2 + 6k + 98 + 30
det(A) = 6k + 130
Now, we set the determinant equal to zero and solve for k:
6k + 130 = 0
6k = -130
k = -130/6
k ≈ -21.67
So, the value of k for which the system does not have a unique solution is approximately -21.67. However, since none of the given answer choices match this value, we can conclude that none of the options provided (A, B, C, D) are correct.
To determine the value of k for which the given system of equations does not have a unique solution, we need to check if the system is consistent or inconsistent using the determinant of the coefficient matrix.
The system can be written in matrix form as Ax = b, where:
A = [[-1, 7, 2], [2, 1, k], [3, -6, -2]]
x = [x, y, z]
b = [6, -11, -25]
If the determinant of matrix A is zero, the system does not have a unique solution. Otherwise, it has a unique solution.
So, let's calculate the determinant of A for different values of k and check which one makes it zero.
Determinant of A = |-1 7 2|
| 2 1 k|
| 3 -6 -2|
To calculate the determinant using the cofactor expansion method, we'll expand along the first row:
Determinant of A = -1 * |-6 -2|
|-6 -2| - 7 * |-2 3|
|-3 -2| + 2 * |-6 3|
= -1 * (-6 * (-2) - (-2) * (-6)) - 7 * (-2 * 3 - (-2) * (-3)) + 2 * (-6 * 3 - (-3) * (-6))
= -1 * (12 - 12) - 7 * (-6 + 6) + 2 * (-18 + 18)
= -1 * 0 - 7 * 0 + 2 * 0
= 0
The determinant of A is zero for all values of k. This means that the system does not have a unique solution for any value of k.
Therefore, the correct answer is: None of the given options (A, B, C, or D)