Answer: To solve the linear system using determinants, we can write the system of equations in matrix form:
| 2 3 | | x | | 6 |
| -8 -3 | | y | = | 12 |
The determinant of the coefficient matrix is calculated as follows:
D = | 2 3 |
| -8 -3 |
The determinant of a 2x2 matrix is calculated by taking the product of the main diagonal elements and subtracting the product of the off-diagonal elements:
D = (2 * -3) - (3 * -8)
D = -6 + 24
D = 18
Now, we will find the determinant of the x matrix, which is obtained by replacing the x column in the coefficient matrix with the constants:
Dx = | 6 3 |
| 12 -3 |
Dx = (6 * -3) - (3 * 12)
Dx = -18 - 36
Dx = -54
Next, we find the determinant of the y matrix, which is obtained by replacing the y column in the coefficient matrix with the constants:
Dy = | 2 6 |
| -8 12 |
Dy = (2 * 12) - (6 * -8)
Dy = 24 + 48
Dy = 72
Finally, we can solve for x and y using the determinants:
x = Dx / D
x = -54 / 18
x = -3
y = Dy / D
y = 72 / 18
y = 4
Therefore, the solution to the given linear system is x = -3 and y = 4.