Let's solve each part of the question step by step.

a. Line passing through $(-2, 5)$ and parallel to $3x - 4y = 6$ in slope-intercept form

First, find the slope of the given line $3x - 4y = 6$.

1. Rewrite the equation in slope-intercept form ($y = mx + b$): $3x - 4y = 6 \implies -4y = -3x + 6 \implies y = \frac{3}{4}x - \frac{3}{2}$ The slope $m$ is $\frac{3}{4}$.

2. Since parallel lines have the same slope, the slope of the required line is also $\frac{3}{4}$.

3. Use the point-slope form to find the equation of the line: $y - y_1 = m(x - x_1)$ Substituting $m = \frac{3}{4}$, $x_1 = -2$, and $y_1 = 5$: $y - 5 = \frac{3}{4}(x + 2)$

4. Simplify to slope-intercept form: $y - 5 = \frac{3}{4}x + \frac{3}{2} \implies y = \frac{3}{4}x + \frac{3}{2} + 5 \implies y = \frac{3}{4}x + \frac{13}{2}$

b. Line passing through $(4, -3)$ and perpendicular to $-3x - 2y = 8$ in point-slope form

First, find the slope of the given line $-3x - 2y = 8$.

1. Rewrite the equation in slope-intercept form: $-3x - 2y = 8 \implies -2y = 3x + 8 \implies y = -\frac{3}{2}x - 4$ The slope $m$ is $-\frac{3}{2}$.

2. The slope of the line perpendicular to this is the negative reciprocal: $m = \frac{2}{3}$

3. Use the point-slope form: $y - y_1 = m(x - x_1)$ Substituting $m = \frac{2}{3}$, $x_1 = 4$, and $y_1 = -3$: $y + 3 = \frac{2}{3}(x - 4)$

c. Line parallel to $y = 8$ and passing through $(-1, -7)$

1. A line parallel to $y = 8$ is a horizontal line with the same slope ($m = 0$).
2. Therefore, the equation of the line passing through $(-1, -7)$ is: $y = -7$

d. Line perpendicular to $x = 2$ and passing through $(-4, 2)$

1. A line perpendicular to $x = 2$ (a vertical line) is a horizontal line.
2. Therefore, the equation of the line passing through $(-4, 2)$ is: $y = 2$

Would you like more details or have any questions?

Here are 8 relative questions expanding on this information:

1. How do you determine the slope of a line from its equation?
2. What is the significance of the slope in the context of parallel and perpendicular lines?
3. Can you explain how to convert between different forms of linear equations (point-slope, slope-intercept, general)?
4. How do you graph a line given in point-slope form?
5. What is the process for finding the equation of a line given two points?
6. How do you find the distance between a point and a line?
7. Can you describe how to check if two lines are parallel or perpendicular?
8. How do you find the midpoint of a line segment connecting two points?

Tip: When dealing with parallel and perpendicular lines, always remember that parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.