Question Find the slope of a line perpendicular to the line whose equation is \( 3 x+18 y=108 \). Fully simplify your answer.

To find the slope of a line perpendicular to the given line \( 3x + 18y = 108 \), follow these steps: ### 1. Find the Slope of the Given Line First, convert the equation \( 3x + 18y = 108 \) to **slope-intercept form** \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. \[ \begin{align*} 3x + 18y &= 108 \\ 18y &= -3x + 108 \quad &\text{(Subtract \( 3x \) from both sides)} \\ y &= \frac{-3}{18}x + \frac{108}{18} \quad &\text{(Divide both sides by 18)} \\ y &= -\frac{1}{6}x + 6 \end{align*} \] **Slope of the given line (\( m \)):** \( -\frac{1}{6} \) ### 2. Determine the Slope of the Perpendicular Line Perpendicular lines have slopes that are **negative reciprocals** of each other. This means if one line has a slope of \( m \), the perpendicular line will have a slope of \( -\frac{1}{m} \). \[ \text{Slope of perpendicular line} = -\frac{1}{m} = -\frac{1}{-\frac{1}{6}} = 6 \] ### **Final Answer** The slope of a line perpendicular to \( 3x + 18y = 108 \) is **6**.