Use Fausta's method to solve these simultaneous equations. \( \begin{array}{ll}\text { i } x+2 y=8 \text { and } & \text { Tip } \\ 5 x=2 y+4 \\ \text { ii } \\ 3 x+2 y=28 \text { and } \\ 3 y=3 x+12\end{array} \)

Certainly! Let's solve each set of simultaneous equations using **Fausta's method**, which typically involves systematically applying elimination or substitution to find the values of the variables. --- ### **Problem i** **Given Equations:** 1. \( x + 2y = 8 \)    \[ \text{(Equation 1)} \] 2. \( 5x = 2y + 4 \)   \[ \text{(Equation 2)} \] **Step 1: Rearrange Equation 2** First, let's rearrange Equation 2 to align it with the standard form: \[ 5x - 2y = 4 \] Now, our system looks like: \[ \begin{cases} x + 2y = 8 \quad &\text{(Equation 1)} \\ 5x - 2y = 4 \quad &\text{(Equation 2)} \end{cases} \] **Step 2: Eliminate \( y \) by Adding Equations** Add Equation 1 and Equation 2 to eliminate \( y \): \[ (x + 5x) + (2y - 2y) = 8 + 4 \] \[ 6x = 12 \] \[ x = \frac{12}{6} = 2 \] **Step 3: Substitute \( x \) into Equation 1 to Find \( y \)** \[ 2 + 2y = 8 \] \[ 2y = 8 - 2 = 6 \] \[ y = \frac{6}{2} = 3 \] **Solution for Problem i:** \[ x = 2,\quad y = 3 \] --- ### **Problem ii** **Given Equations:** 1. \( 3x + 2y = 28 \)    \[ \text{(Equation 1)} \] 2. \( 3y = 3x + 12 \)   \[ \text{(Equation 2)} \] **Step 1: Rearrange Equation 2** First, let's rearrange Equation 2 to align it with the standard form: \[ 3y - 3x = 12 \] \[ -3x + 3y = 12 \] Now, our system looks like: \[ \begin{cases} 3x + 2y = 28 \quad &\text{(Equation 1)} \\ -3x + 3y = 12 \quad &\text{(Equation 2)} \end{cases} \] **Step 2: Eliminate \( x \) by Adding Equations** Add Equation 1 and Equation 2 to eliminate \( x \): \[ (3x - 3x) + (2y + 3y) = 28 + 12 \] \[ 0x + 5y = 40 \] \[ 5y = 40 \] \[ y = \frac{40}{5} = 8 \] **Step 3: Substitute \( y \) into Equation 1 to Find \( x \)** \[ 3x + 2(8) = 28 \] \[ 3x + 16 = 28 \] \[ 3x = 28 - 16 = 12 \] \[ x = \frac{12}{3} = 4 \] **Solution for Problem ii:** \[ x = 4,\quad y = 8 \] --- ### **Final Answers:** - **Problem i:** \( x = 2 \), \( y = 3 \) - **Problem ii:** \( x = 4 \), \( y = 8 \)