Final answer:
The determinant of the coefficient matrix for the given system is 0. Since Cramer's rule requires a non-zero determinant, it cannot be applied to find a unique solution for this system. The equations are dependent, hence the system has infinitely many solutions or no solution.
Step-by-step explanation:
To solve the given system of equations using Cramer's rule, we must find the determinant of the coefficient matrix and the determinants of matrices formed by replacing each column with the constants from the right side of the equations.
The system is:
3x - 2y = 4
6x - 4y = 0
The coefficient matrix and its determinant (D) is:
| 3 -2 |
| 6 -4 |
D = (3 * -4) - (6 * -2) = -12 - (-12) = 0
For x (Dx), replace the x column with the constants:
| 4 -2 |
| 0 -4 |
Dx = (4 * -4) - (0 * -2) = -16
For y (Dy), replace the y column with the constants:
| 3 4 |
| 6 0 |
Dy = (3 * 0) - (4 * 6) = -24
Since the determinant D of the coefficient matrix is 0, we cannot use Cramer's rule to find a unique solution because the system is either dependent or inconsistent. In this case, the equations are multiples of one another, indicating a dependent system with infinitely many solutions or no solution.