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

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To find the slope of a line perpendicular to the given line $$ 3x - 6y = -72 $$, follow these steps:

1. **Rearrange the Equation to Slope-Intercept Form ($$ y = mx + b $$)**:

Start with the given equation:
   $$
   3x - 6y = -72
   $$
   
   Solve for $$ y $$:
   $$
   -6y = -3x - 72
   $$
   $$
   y = \frac{-3x - 72}{-6}
   $$
   $$
   y = \frac{3x}{6} + \frac{72}{6}
   $$
   $$
   y = \frac{1}{2}x + 12
   $$
   
   So, the slope ($$ m $$) of the given line is **$$ \frac{1}{2} $$**.

2. **Determine the Slope of the Perpendicular Line**:

Perpendicular lines have slopes that are **negative reciprocals** of each other. This means if the slope of one line is $$ m $$, the slope of a perpendicular line is $$ -\frac{1}{m} $$.

Given the slope $$ m = \frac{1}{2} $$, the slope of the perpendicular line ($$ m_{\text{perpendicular}} $$) is:
   $$
   m_{\text{perpendicular}} = -\frac{1}{\frac{1}{2}} = -2
   $$
   
3. **Final Answer**:

The slope of a line perpendicular to the given line is **$$ -2 $$**.
The slope of a line perpendicular to $$ 3x - 6y = -72 $$ is **-2**.

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

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