1. **Problem Statement:** Solve the system of linear equations using Cramer's Rule: $$5x + 2y - 2 = 0$$ $$2x + 3y + 8 = 0$$ 2. **Rewrite equations in standard form:** $$5x + 2y = 2$$ $$2x + 3y = -8$$ 3. **Formula for Cramer's Rule:** For system: $$ax + by = e$$ $$cx + dy = f$$ The solutions are: $$x = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$$ $$y = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$$ 4. **Identify coefficients:** $$a=5, b=2, e=2$$ $$c=2, d=3, f=-8$$ 5. **Calculate determinant of coefficient matrix:** $$D = \begin{vmatrix} 5 & 2 \\ 2 & 3 \end{vmatrix} = (5)(3) - (2)(2) = 15 - 4 = 11$$ 6. **Calculate determinant for x:** $$D_x = \begin{vmatrix} 2 & 2 \\ -8 & 3 \end{vmatrix} = (2)(3) - (2)(-8) = 6 + 16 = 22$$ 7. **Calculate determinant for y:** $$D_y = \begin{vmatrix} 5 & 2 \\ 2 & -8 \end{vmatrix} = (5)(-8) - (2)(2) = -40 - 4 = -44$$ 8. **Calculate values of x and y:** $$x = \frac{D_x}{D} = \frac{22}{11} = 2$$ $$y = \frac{D_y}{D} = \frac{-44}{11} = -4$$ **Final answer:** $$x = 2, \quad y = -4$$