Find the slope of a line perpendicular to the line whose equation is \( 3 x-6 y=-72 \). Fully simplify your answer.

To find the slope of a line perpendicular to the given line, we first need to determine the slope of the original line from its equation \( 3x - 6y = -72 \). We can rewrite this in slope-intercept form (y = mx + b). 1. Rearranging the equation: \( -6y = -3x - 72 \) \( y = \frac{1}{2}x + 12 \) From this, we can see that the slope (m) of the original line is \( \frac{1}{2} \). 2. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope of the perpendicular line is: \[ m_{\text{perpendicular}} = -\frac{1}{\left(\frac{1}{2}\right)} = -2 \] Finally, the slope of the line perpendicular to the given line is \(-2\).