Find the value of 'k' for which the system of equations 4x + 6y = 7 and 6x + (k + 4)y - 21 = 0 has a unique solution.

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Given the system of equations:
1) 4x + 6y = 7
2) 6x + (k + 4)y - 21 = 0

Rewrite the second equation as:
6x + (k + 4)y = 21

For the system to have a unique solution, the determinant of the coefficient matrix must be non-zero.

Coefficient matrix:  
| 4    6   |  
| 6  k+4 |

Determinant (D) = 4(k + 4) - 6*6 = 4k + 16 - 36 = 4k - 20

For unique solution, D ≠ 0
=> 4k - 20 ≠ 0
=> 4k ≠ 20
=> k ≠ 5

Therefore, the system has a unique solution for all values of k except k = 5.

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