Final answer:
The system of equations that has exactly one solution is option A: {3x - y = 8, 6x - 2y = 16}.
Step-by-step explanation:
The system of equations that has exactly one solution is option A: {3x - y = 8, 6x - 2y = 16}.
To determine if a system of equations has exactly one solution, we can use the determinant of the coefficient matrix.
If the determinant is nonzero, then the system has exactly one solution. If the determinant is zero, then the system has either no solution or infinitely many solutions.
In this case, the determinant of the coefficient matrix is 6 - 6 = 0, which means that option B: {-6x + y = 2, y = 6x + 2} has either no solution or infinitely many solutions.
The determinant of the coefficient matrix for option C: {y = 4x + 9, y = 4x - 9} is also zero, so it has either no solution or infinitely many solutions.
Finally, the determinant of the coefficient matrix for option D: {3x + 2y = 4, x - y = 3} is 3 - (-2) = 5, which is nonzero, indicating that it has exactly one solution.