Elimination was used to solve a system of equations. One of the intermediate steps led to the equation \( 3 x=18 \). Which of the following systems could have led to this equatio \[ 4 x+y=20 \] \( x-y=2 \) \( 2 x+y=24 \) \( -x-y=6 \) \[ x+y=4 \] \[ x-2 y=10 \] \[ 3 x+y=18 \] \( -3 x-y=-18 \)

To determine which system of equations could have led to the intermediate step \( 3x = 18 \), we can solve each system of equations and check if we can derive \( 3x = 18 \) from any of them. ### Step 1: Solve each system of equations 1. **System 1:** \[ \begin{align*} 4x + y &= 20 \quad (1) \\ x - y &= 2 \quad (2) \end{align*} \] From equation (2), we can express \( y \) in terms of \( x \): \[ y = x - 2 \] Substitute \( y \) into equation (1): \[ 4x + (x - 2) = 20 \\ 5x - 2 = 20 \\ 5x = 22 \\ x = \frac{22}{5} \] This does not lead to \( 3x = 18 \). 2. **System 2:** \[ \begin{align*} 2x + y &= 24 \quad (1) \\ -x - y &= 6 \quad (2) \end{align*} \] From equation (2), we can express \( y \): \[ y = -x - 6 \] Substitute \( y \) into equation (1): \[ 2x + (-x - 6) = 24 \\ 2x - x - 6 = 24 \\ x - 6 = 24 \\ x = 30 \] This does not lead to \( 3x = 18 \). 3. **System 3:** \[ \begin{align*} x + y &= 4 \quad (1) \\ x - 2y &= 10 \quad (2) \end{align*} \] From equation (1), we can express \( y \): \[ y = 4 - x \] Substitute \( y \) into equation (2): \[ x - 2(4 - x) = 10 \\ x - 8 + 2x = 10 \\ 3x - 8 = 10 \\ 3x = 18 \] This leads to \( 3x = 18 \). 4. **System 4:** \[ \begin{align*} 3x + y &= 18 \quad (1) \\ -3x - y &= -18 \quad (2) \end{align*} \] From equation (1), we already have \( 3x + y = 18 \), which is the same as the intermediate step. Thus, this system also leads to \( 3x = 18 \). ### Conclusion The systems that could have led to the equation \( 3x = 18 \) are: - System 3: \[ x + y = 4 \\ x - 2y = 10 \] - System 4: \[ 3x + y = 18 \\ -3x - y = -18 \] Thus, the answer is **System 3 and System 4**.