Perpendicular Line 852579

1. **State the problem:** Find the equation of the line perpendicular to $y = -\frac{2}{3} x$ that passes through the point $(4,-8)$. 2. **Recall the slope of the given line:** The given line is $y = -\frac{2}{3} x$, so its slope is $m_1 = -\frac{2}{3}$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. $$m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2}$$ 4. **Use the point-slope form:** The equation of a line with slope $m$ passing through point $(x_1,y_1)$ is $$y - y_1 = m(x - x_1)$$ Substitute $m = \frac{3}{2}$ and point $(4,-8)$: $$y - (-8) = \frac{3}{2}(x - 4)$$ 5. **Simplify the equation:** $$y + 8 = \frac{3}{2}x - \frac{3}{2} \times 4$$ $$y + 8 = \frac{3}{2}x - 6$$ 6. **Isolate $y$ to get slope-intercept form:** $$y = \frac{3}{2}x - 6 - 8$$ $$y = \frac{3}{2}x - 14$$ 7. **Final answer:** The equation of the line perpendicular to $y = -\frac{2}{3} x$ and passing through $(4,-8)$ is $$y = \frac{3}{2}x - 14$$