1. **State the problem:** Solve the system of equations using Cramer's rule: $$3x - 2y = 4$$ $$2x + y = 7$$ 2. **Recall Cramer's rule:** For a system $$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$ The solutions are given by: $$x = \frac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}, \quad y = \frac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$$ 3. **Identify coefficients:** $$a_1 = 3, b_1 = -2, c_1 = 4$$ $$a_2 = 2, b_2 = 1, c_2 = 7$$ 4. **Calculate the determinant of the coefficient matrix:** $$D = \begin{vmatrix} 3 & -2 \\ 2 & 1 \end{vmatrix} = (3)(1) - (-2)(2) = 3 + 4 = 7$$ 5. **Calculate determinant for $x$:** $$D_x = \begin{vmatrix} 4 & -2 \\ 7 & 1 \end{vmatrix} = (4)(1) - (-2)(7) = 4 + 14 = 18$$ 6. **Calculate determinant for $y$:** $$D_y = \begin{vmatrix} 3 & 4 \\ 2 & 7 \end{vmatrix} = (3)(7) - (4)(2) = 21 - 8 = 13$$ 7. **Find $x$ and $y$:** $$x = \frac{D_x}{D} = \frac{18}{7}$$ $$y = \frac{D_y}{D} = \frac{13}{7}$$ **Final answer:** $$x = \frac{18}{7}, \quad y = \frac{13}{7}$$