1. **State the problem:** Find the equation of the line perpendicular to the line $4x - 3y = -8$ that passes through the point $(3, -2)$. 2. **Convert the given line to slope-intercept form:** $$4x - 3y = -8 \implies -3y = -4x - 8 \implies y = \frac{4}{3}x + \frac{8}{3}$$ The slope of the given line is $\frac{4}{3}$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of $\frac{4}{3}$ is $-\frac{3}{4}$, not $-\frac{4}{3}$. 4. **Use point-slope form with the correct slope:** Point-slope form is: $$y - y_1 = m(x - x_1)$$ Substitute $m = -\frac{3}{4}$ and point $(3, -2)$: $$y - (-2) = -\frac{3}{4}(x - 3)$$ which simplifies to: $$y + 2 = -\frac{3}{4}(x - 3)$$ 5. **Simplify the equation:** $$y + 2 = -\frac{3}{4}x + \frac{9}{4}$$ Subtract 2 from both sides: $$y = -\frac{3}{4}x + \frac{9}{4} - 2$$ Write 2 as $\frac{8}{4}$: $$y = -\frac{3}{4}x + \frac{9}{4} - \frac{8}{4}$$ Simplify: $$y = -\frac{3}{4}x + \frac{1}{4}$$ **Final answer:** $$y = -\frac{3}{4}x + \frac{1}{4}$$