1. **State the problem:** Find the equation of the line passing through point $(2, -5)$ and perpendicular to the line given by $3x - 6y = 24$. 2. **Rewrite the given line in slope-intercept form:** $$3x - 6y = 24 \implies -6y = -3x + 24 \implies y = \frac{1}{2}x - 4$$ The slope of the given line is $m = \frac{1}{2}$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another with slope $m$ is the negative reciprocal: $$m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{1}{2}} = -2$$ 4. **Use point-slope form to find the equation of the perpendicular line:** Point-slope form is: $$y - y_1 = m(x - x_1)$$ Using point $(2, -5)$ and slope $-2$: $$y - (-5) = -2(x - 2)$$ $$y + 5 = -2x + 4$$ $$y = -2x + 4 - 5$$ $$y = -2x - 1$$ 5. **Final answer:** The equation of the line perpendicular to $3x - 6y = 24$ and passing through $(2, -5)$ is: $$\boxed{y = -2x - 1}$$ 6. **Graphing both lines:** - Original line: $y = \frac{1}{2}x - 4$ - Perpendicular line: $y = -2x - 1$