Solve using elimination. \( \begin{array}{l}8 x+9 y=17 \\ -3 x-2 y=-5\end{array} \)

To solve the system of equations using the **elimination method**, follow these steps: \[ \begin{cases} 8x + 9y = 17 \quad \text{(Equation 1)} \\ -3x - 2y = -5 \quad \text{(Equation 2)} \end{cases} \] **Step 1: Align the Equations** First, write the equations clearly: 1. \( 8x + 9y = 17 \) 2. \( -3x - 2y = -5 \) **Step 2: Make the Coefficients of One Variable Opposite** We'll eliminate \( y \) by making the coefficients of \( y \) in both equations opposites. To do this, find the Least Common Multiple (LCM) of 9 and 2, which is 18. - Multiply **Equation 1** by **2**: \[ 2 \times (8x + 9y) = 2 \times 17 \\ 16x + 18y = 34 \quad \text{(Equation 3)} \] - Multiply **Equation 2** by **9**: \[ 9 \times (-3x - 2y) = 9 \times (-5) \\ -27x - 18y = -45 \quad \text{(Equation 4)} \] **Step 3: Add the Modified Equations** Add **Equation 3** and **Equation 4** to eliminate \( y \): \[ \begin{align*} 16x + 18y &= 34 \\ -27x - 18y &= -45 \\ \hline (16x - 27x) + (18y - 18y) &= 34 - 45 \\ -11x &= -11 \\ \end{align*} \] **Step 4: Solve for \( x \)** \[ -11x = -11 \\ x = \frac{-11}{-11} \\ x = 1 \] **Step 5: Substitute \( x \) into One of the Original Equations** Using **Equation 1**: \[ 8x + 9y = 17 \\ 8(1) + 9y = 17 \\ 8 + 9y = 17 \\ 9y = 17 - 8 \\ 9y = 9 \\ y = \frac{9}{9} \\ y = 1 \] **Solution:** \[ x = 1 \\ y = 1 \]