Question: explain from point 3 to point 4 in detail breaking it down extra further 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 $6 = 4y + 3x$. 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: $m_1 = -\frac{3}{4}$. 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: $m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{3}{4}} = \frac{4}{3}$. 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: $y + 4 = \frac{4}{3}x - \frac{4}{3} \times 6$. $y + 4 = \frac{4}{3}x - 8$. Isolate $y$ to get slope-intercept form: $y = \frac{4}{3}x - 8 - 4$. $y = \frac{4}{3}x - 12$. Final answer: $y = \frac{4}{3}x - 12$. 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.
1. **State the problem:** We need to find the equation of a line that passes 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.
3. **Start with the given equation:**
$$6 = 4y + 3x$$
4. **Subtract $3x$ from both sides:**
$$6 - 3x = 4y$$
5. **Divide both sides by 4 to isolate $y$:**
$$y = \frac{6 - 3x}{4} = \frac{6}{4} - \frac{3}{4}x = \frac{3}{2} - \frac{3}{4}x$$
6. **Rewrite to standard slope-intercept form:**
$$y = -\frac{3}{4}x + \frac{3}{2}$$
7. **Identify the slope of the given line:**
$$m_1 = -\frac{3}{4}$$
8. **Find the slope of the perpendicular line:**
The slope of a line perpendicular to another is the negative reciprocal of the original slope.
9. **Calculate the negative reciprocal:**
$$m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{3}{4}} = \frac{4}{3}$$
10. **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 $m = \frac{4}{3}$.
11. **Substitute the values:**
$$y - (-4) = \frac{4}{3}(x - 6)$$
12. **Simplify the left side:**
$$y + 4 = \frac{4}{3}x - \frac{4}{3} \times 6$$
13. **Calculate the multiplication:**
$$y + 4 = \frac{4}{3}x - 8$$
14. **Isolate $y$ to get slope-intercept form:**
$$y = \frac{4}{3}x - 8 - 4$$
15. **Simplify the constants:**
$$y = \frac{4}{3}x - 12$$
16. **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.