1. **Problem Statement:** Solve a system of two linear equations using three methods: Cramer's Rule, Inversion Method, and Gauss Elimination Method. Provide two examples for each method. --- ### Cramer's Rule 2. **Method Explanation:** For a system $\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$, Cramer's Rule uses determinants: $$D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1b_2 - a_2b_1$$ $$D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} = c_1b_2 - c_2b_1$$ $$D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} = a_1c_2 - a_2c_1$$ Solution: $$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}$$ 3. **Example 1:** Solve $\begin{cases} 2x + 3y = 5 \\ 4x - y = 1 \end{cases}$ Calculate determinants: $$D = 2 \times (-1) - 4 \times 3 = -2 - 12 = -14$$ $$D_x = 5 \times (-1) - 1 \times 3 = -5 - 3 = -8$$ $$D_y = 2 \times 1 - 4 \times 5 = 2 - 20 = -18$$ Solution: $$x = \frac{-8}{-14} = \frac{4}{7}, \quad y = \frac{-18}{-14} = \frac{9}{7}$$ 4. **Example 2:** Solve $\begin{cases} x - 2y = 3 \\ 3x + y = 7 \end{cases}$ Calculate determinants: $$D = 1 \times 1 - 3 \times (-2) = 1 + 6 = 7$$ $$D_x = 3 \times 1 - 7 \times (-2) = 3 + 14 = 17$$ $$D_y = 1 \times 7 - 3 \times 3 = 7 - 9 = -2$$ Solution: $$x = \frac{17}{7}, \quad y = \frac{-2}{7}$$ --- ### Inversion Method 5. **Method Explanation:** Write system as $AX = B$ where $$A = \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}$$ If $A^{-1}$ exists, then $$X = A^{-1}B$$ For 2x2 matrix, $$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} b_2 & -b_1 \\ -a_2 & a_1 \end{bmatrix}$$ 6. **Example 1:** Solve $\begin{cases} 1x + 2y = 5 \\ 3x + 4y = 11 \end{cases}$ Calculate determinant: $$\det(A) = 1 \times 4 - 3 \times 2 = 4 - 6 = -2$$ Calculate inverse: $$A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}$$ Multiply: $$X = A^{-1}B = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \begin{bmatrix} 5 \\ 11 \end{bmatrix} = \begin{bmatrix} -2 \times 5 + 1 \times 11 \\ 1.5 \times 5 - 0.5 \times 11 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$$ 7. **Example 2:** Solve $\begin{cases} 4x + y = 9 \\ 2x + 3y = 13 \end{cases}$ Calculate determinant: $$\det(A) = 4 \times 3 - 2 \times 1 = 12 - 2 = 10$$ Calculate inverse: $$A^{-1} = \frac{1}{10} \begin{bmatrix} 3 & -1 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.3 & -0.1 \\ -0.2 & 0.4 \end{bmatrix}$$ Multiply: $$X = A^{-1}B = \begin{bmatrix} 0.3 & -0.1 \\ -0.2 & 0.4 \end{bmatrix} \begin{bmatrix} 9 \\ 13 \end{bmatrix} = \begin{bmatrix} 0.3 \times 9 - 0.1 \times 13 \\ -0.2 \times 9 + 0.4 \times 13 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$$ --- ### Gauss Elimination Method 8. **Method Explanation:** Transform system to upper triangular form and solve by back substitution. 9. **Example 1:** Solve $\begin{cases} 2x + 3y = 8 \\ 4x + y = 10 \end{cases}$ Multiply first equation by 2: $$4x + 6y = 16$$ Subtract second equation: $$(4x + 6y) - (4x + y) = 16 - 10 \Rightarrow 5y = 6 \Rightarrow y = \frac{6}{5}$$ Substitute back: $$2x + 3 \times \frac{6}{5} = 8 \Rightarrow 2x + \frac{18}{5} = 8 \Rightarrow 2x = 8 - \frac{18}{5} = \frac{40}{5} - \frac{18}{5} = \frac{22}{5} \Rightarrow x = \frac{11}{5}$$ 10. **Example 2:** Solve $\begin{cases} x + 2y = 7 \\ 3x + 4y = 18 \end{cases}$ Multiply first equation by 3: $$3x + 6y = 21$$ Subtract second equation: $$(3x + 6y) - (3x + 4y) = 21 - 18 \Rightarrow 2y = 3 \Rightarrow y = \frac{3}{2}$$ Substitute back: $$x + 2 \times \frac{3}{2} = 7 \Rightarrow x + 3 = 7 \Rightarrow x = 4$$ --- **Final answers:** - Cramer's Rule examples: $x=\frac{4}{7}, y=\frac{9}{7}$ and $x=\frac{17}{7}, y=\frac{-2}{7}$ - Inversion Method examples: $x=1, y=2$ and $x=2, y=3$ - Gauss Elimination examples: $x=\frac{11}{5}, y=\frac{6}{5}$ and $x=4, y=\frac{3}{2}$