1. **State the problem:** Find the equation of the line passing through the point $(-6, 3)$ and perpendicular to the line given by $6x - 5y = 5$. 2. **Rewrite the given line in slope-intercept form:** $$6x - 5y = 5$$ Subtract $6x$ from both sides: $$-5y = -6x + 5$$ Divide both sides by $-5$: $$y = \cancel{\frac{-6}{-5}}{\frac{6}{5}}x - \cancel{\frac{5}{-5}}{-1}$$ So the slope of the given line is $m = \frac{6}{5}$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. $$m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{6}{5}} = -\frac{5}{6}$$ 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 $(-6, 3)$ and slope $-\frac{5}{6}$: $$y - 3 = -\frac{5}{6}(x - (-6)) = -\frac{5}{6}(x + 6)$$ 5. **Simplify the equation:** $$y - 3 = -\frac{5}{6}x - \frac{5}{6} \times 6$$ $$y - 3 = -\frac{5}{6}x - 5$$ Add 3 to both sides: $$y = -\frac{5}{6}x - 5 + 3$$ $$y = -\frac{5}{6}x - 2$$ **Final answer:** $$\boxed{y = -\frac{5}{6}x - 2}$$