1. **State the problem:** Solve the system of equations using determinants (Cramer's Rule): $$4x - 4y - 8 = 0$$ $$3x + 5 = 2y$$ 2. **Rewrite the equations in standard form:** Equation 1: $$4x - 4y = 8$$ Equation 2: $$3x - 2y = -5$$ (moved all terms to one side) 3. **Write the coefficient matrix and constants vector:** Coefficient matrix $$A = \begin{bmatrix}4 & -4 \\ 3 & -2\end{bmatrix}$$ Constants vector $$B = \begin{bmatrix}8 \\ -5\end{bmatrix}$$ 4. **Calculate the determinant of matrix A:** $$D = \det(A) = (4)(-2) - (-4)(3) = -8 + 12 = 4$$ Since $$D \neq 0$$, the system has a unique solution. 5. **Calculate determinant $$D_x$$ by replacing the first column of A with B:** $$D_x = \det\begin{bmatrix}8 & -4 \\ -5 & -2\end{bmatrix} = (8)(-2) - (-4)(-5) = -16 - 20 = -36$$ 6. **Calculate determinant $$D_y$$ by replacing the second column of A with B:** $$D_y = \det\begin{bmatrix}4 & 8 \\ 3 & -5\end{bmatrix} = (4)(-5) - (8)(3) = -20 - 24 = -44$$ 7. **Find the values of $$x$$ and $$y$$ using Cramer's Rule:** $$x = \frac{D_x}{D} = \frac{-36}{4} = -9$$ $$y = \frac{D_y}{D} = \frac{-44}{4} = -11$$ **Final answer:** $$x = -9$$ and $$y = -11$$