Perpendicular Line 1Bb383

1. **State the problem:** We need to find the equation of a line that goes through the point $(6, -4)$ and is perpendicular to the line given by the equation $$6 = 4y + 3x$$. 2. **Rewrite the given line in slope-intercept form:** The slope-intercept form is $$y = mx + b$$ where $m$ is the slope. Start with: $$6 = 4y + 3x$$ Subtract $3x$ from both sides: $$6 - 3x = 4y$$ Divide both sides by 4: $$y = \frac{6 - 3x}{4} = \frac{6}{4} - \frac{3}{4}x = \frac{3}{2} - \frac{3}{4}x$$ Rewrite to standard slope-intercept form: $$y = -\frac{3}{4}x + \frac{3}{2}$$ So the slope of the given line is: $$m_1 = -\frac{3}{4}$$ 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. Calculate the negative reciprocal: $$m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{3}{4}} = \frac{4}{3}$$ 4. **Use point-slope form to find the equation of the new line:** The point-slope form is: $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1) = (6, -4)$ and $m = \frac{4}{3}$. Substitute values: $$y - (-4) = \frac{4}{3}(x - 6)$$ Simplify: $$y + 4 = \frac{4}{3}x - \frac{4}{3} \times 6$$ $$y + 4 = \frac{4}{3}x - 8$$ 5. **Isolate $y$ to get slope-intercept form:** $$y = \frac{4}{3}x - 8 - 4$$ $$y = \frac{4}{3}x - 12$$ **Final answer:** $$\boxed{y = \frac{4}{3}x - 12}$$ This is the equation of the line passing through $(6, -4)$ and perpendicular to the given line. **Explanation for dummies:** - First, find the slope of the original line by rewriting it nicely. - Then, flip that slope upside down and change its sign to get the perpendicular slope. - Use the point you have and the new slope to write the equation. - Simplify to get the final answer.