1. **State the problem:** Solve the system of equations: $$\begin{cases} x + y = 13 \\ x - 2y = 5 \end{cases}$$ 2. **Formula and rules:** To solve a system of two linear equations, we can use Cramer's rule. The determinant of the coefficient matrix is: $$\Delta = \begin{vmatrix} 1 & 1 \\ 1 & -2 \end{vmatrix} = (1)(-2) - (1)(1) = -2 - 1 = -3$$ 3. **Calculate determinant $\Delta_x$:** Replace the first column with constants: $$\Delta_x = \begin{vmatrix} 13 & 1 \\ 5 & -2 \end{vmatrix} = (13)(-2) - (1)(5) = -26 - 5 = -31$$ 4. **Calculate determinant $\Delta_y$:** Replace the second column with constants: $$\Delta_y = \begin{vmatrix} 1 & 13 \\ 1 & 5 \end{vmatrix} = (1)(5) - (13)(1) = 5 - 13 = -8$$ 5. **Find solutions using Cramer's rule:** $$x = \frac{\Delta_x}{\Delta} = \frac{-31}{-3} = \frac{31}{3}$$ $$y = \frac{\Delta_y}{\Delta} = \frac{-8}{-3} = \frac{8}{3}$$ 6. **Interpretation:** The solution is: $$x = \frac{31}{3} \approx 10.33, \quad y = \frac{8}{3} \approx 2.67$$ This means the values of $x$ and $y$ satisfy both equations in the system.