Which of the following equations represents a line that is perpendicular to \( y=-3 x+6 \) and passes through the point, \( (3,2) \) ? A. \( y=-3 x+1 \) B. \( y=\frac{1}{3} x+1 \) C. \( y=-\frac{1}{3} x+1 \) D. \( y=\frac{1}{3} x+3 \)

To determine which equation represents a line perpendicular to \( y = -3x + 6 \) and passes through the point \( (3, 2) \), follow these steps: 1. **Find the slope of the given line:** The given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. For \( y = -3x + 6 \), the slope \( m_1 = -3 \). 2. **Determine the slope of the perpendicular line:** Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope \( m_2 \) of the perpendicular line is: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{-3} = \frac{1}{3} \] 3. **Use the point-slope form to find the equation of the perpendicular line:** The point-slope form is: \[ y - y_1 = m(x - x_1) \] Plugging in \( m = \frac{1}{3} \) and the point \( (3, 2) \): \[ y - 2 = \frac{1}{3}(x - 3) \] Simplifying: \[ y = \frac{1}{3}x - 1 + 2 \] \[ y = \frac{1}{3}x + 1 \] 4. **Match with the given options:** The equation \( y = \frac{1}{3}x + 1 \) corresponds to option **B**. **Answer:** B. \( y=\frac{1}{3} x+1 \)