1. **Problem:** Solve the system using Cramer's rule: $$\begin{cases} 3x + 2y = 7 \\ 4x - y = 2 \end{cases}$$ 2. **Formula:** For system $$\begin{cases} a_{11}x + a_{12}y = c_1 \\ a_{21}x + a_{22}y = c_2 \end{cases}$$ Determinant of coefficients: $$D = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}$$ Replace first column with constants for $D_x$: $$D_x = \begin{vmatrix} c_1 & a_{12} \\ c_2 & a_{22} \end{vmatrix} = c_1 a_{22} - a_{12} c_2$$ Replace second column with constants for $D_y$: $$D_y = \begin{vmatrix} a_{11} & c_1 \\ a_{21} & c_2 \end{vmatrix} = a_{11} c_2 - c_1 a_{21}$$ Solutions: $$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}$$ 3. **Calculate $D$:** $$D = \begin{vmatrix} 3 & 2 \\ 4 & -1 \end{vmatrix} = 3 \times (-1) - 2 \times 4 = -3 - 8 = -11$$ 4. **Calculate $D_x$:** $$D_x = \begin{vmatrix} 7 & 2 \\ 2 & -1 \end{vmatrix} = 7 \times (-1) - 2 \times 2 = -7 - 4 = -11$$ 5. **Calculate $D_y$:** $$D_y = \begin{vmatrix} 3 & 7 \\ 4 & 2 \end{vmatrix} = 3 \times 2 - 7 \times 4 = 6 - 28 = -22$$ 6. **Find $x_0$ and $y_0$:** $$x_0 = \frac{D_x}{D} = \frac{-11}{-11} = 1$$ $$y_0 = \frac{D_y}{D} = \frac{-22}{-11} = 2$$ 7. **Find $x_0 + y_0$:** $$1 + 2 = 3$$ --- 1. **Problem:** Solve the system: $$\begin{cases} 6x - y = -13 \\ -4x + 3y = 11 \end{cases}$$ 2. **Calculate $D$:** $$D = \begin{vmatrix} 6 & -1 \\ -4 & 3 \end{vmatrix} = 6 \times 3 - (-1) \times (-4) = 18 - 4 = 14$$ 3. **Calculate $D_x$:** $$D_x = \begin{vmatrix} -13 & -1 \\ 11 & 3 \end{vmatrix} = (-13) \times 3 - (-1) \times 11 = -39 + 11 = -28$$ 4. **Calculate $D_y$:** $$D_y = \begin{vmatrix} 6 & -13 \\ -4 & 11 \end{vmatrix} = 6 \times 11 - (-13) \times (-4) = 66 - 52 = 14$$ 5. **Find $x_0$ and $y_0$:** $$x_0 = \frac{D_x}{D} = \frac{-28}{14} = -2$$ $$y_0 = \frac{D_y}{D} = \frac{14}{14} = 1$$ 6. **Find $x_0 - y_0$:** $$-2 - 1 = -3$$ --- 1. **Problem:** Identify the determinant corresponding to the price of 1 pisang goreng from Jessica and Wesley's purchases: Jessica: 3 pisang goreng + 2 donat = 3500 Wesley: 4 pisang goreng + 2 donat = 4000 2. **Coefficient matrix:** $$\begin{bmatrix} 3 & 2 \\ 4 & 2 \end{bmatrix}$$ Constants vector: $$\begin{bmatrix} 3500 \\ 4000 \end{bmatrix}$$ 3. **Cramer's rule for price of pisang goreng (x):** Replace first column with constants: $$D_x = \begin{vmatrix} 3500 & 2 \\ 4000 & 2 \end{vmatrix} = 3500 \times 2 - 2 \times 4000 = 7000 - 8000 = -1000$$ 4. **Coefficient determinant:** $$D = \begin{vmatrix} 3 & 2 \\ 4 & 2 \end{vmatrix} = 3 \times 2 - 2 \times 4 = 6 - 8 = -2$$ 5. **Price of pisang goreng:** $$x = \frac{D_x}{D} = \frac{-1000}{-2} = 500$$ 6. **Matching determinant for $D_x$ is option E:** $$\begin{vmatrix} 3500 & 2 \\ 4000 & 2 \end{vmatrix}$$ --- 1. **Problem:** Solve for $x'$ and $y'$ in terms of $x$ and $y$ given: $$\begin{cases} x = \frac{3}{5} x' - \frac{4}{5} y' \\ y = \frac{4}{5} x' + \frac{3}{5} y' \end{cases}$$ 2. **Rewrite system in matrix form:** $$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{3}{5} & -\frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix}$$ 3. **Find inverse of coefficient matrix:** Determinant: $$D = \frac{3}{5} \times \frac{3}{5} - (-\frac{4}{5}) \times \frac{4}{5} = \frac{9}{25} + \frac{16}{25} = 1$$ Inverse: $$\begin{bmatrix} \frac{3}{5} & -\frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{bmatrix}^{-1} = \frac{1}{1} \begin{bmatrix} \frac{3}{5} & \frac{4}{5} \\ -\frac{4}{5} & \frac{3}{5} \end{bmatrix} = \begin{bmatrix} \frac{3}{5} & \frac{4}{5} \\ -\frac{4}{5} & \frac{3}{5} \end{bmatrix}$$ 4. **Express $x'$ and $y'$:** $$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \frac{3}{5} & \frac{4}{5} \\ -\frac{4}{5} & \frac{3}{5} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{cases} x' = \frac{3}{5} x + \frac{4}{5} y \\ y' = -\frac{4}{5} x + \frac{3}{5} y \end{cases}$$