Solve the system of linear equations: X + y = 6 and 2x - 3y = -2 using Gramer's rule to find the determinants?

To solve the system of linear equations using Cramer's rule, we first need to find the determinants of the coefficient matrix and the matrices obtained by replacing the x-column with the constants and the y-column with the constants.

The coefficient matrix is:
| 1 1 |
| 2 -3 |

The determinant of the coefficient matrix (denoted as D) is:
D = (1)(-3) - (2)(1) = -3 - 2 = -5

Now, we replace the x-column with the constants:
| 6 1 |
| -2 -3 |

The determinant of this matrix (denoted as Dx) is:
Dx = (6)(-3) - (-2)(1) = -18 + 2 = -16

Next, we replace the y-column with the constants:
| 1 6 |
| 2 -2 |

The determinant of this matrix (denoted as Dy) is:
Dy = (1)(-2) - (2)(6) = -2 - 12 = -14

Now, we can find the values of x and y using Cramer's rule:
x = Dx / D = -16 / -5 = 16/5
y = Dy / D = -14 / -5 = 14/5

Therefore, the solution to the system of linear equations is x = 16/5 and y = 14/5.