https://file.aiquickdraw.com/mathbot/file/1726451382592l0wdcktz.jpg

Let's break down and solve the two parts of this problem:

### Given Data:
- **Point**: \((-9, 3)\)
- **Line**: \(x + 8 = 0\)

This equation represents a vertical line where \(x = -8\).

### Part (a): 
Find the equation of the line **parallel** to the given line and passing through the point \((-9, 3)\).

- A line parallel to \(x = -8\) will also be a vertical line. So, its equation will have the form \(x = \text{constant}\).
- Since the line must pass through the point \((-9, 3)\), the equation of the line is:
  \[
  x = -9
  \]

### Part (b): 
Find the equation of the line **perpendicular** to the given line and passing through the point \((-9, 3)\).

- A line perpendicular to a vertical line is a horizontal line. Horizontal lines have the form \(y = \text{constant}\).
- Since this line must pass through the point \((-9, 3)\), the equation is:
  \[
  y = 3
  \]

### Final Answers:
- (a) The equation of the line parallel to \(x + 8 = 0\) through \((-9, 3)\) is \(x = -9\).
- (b) The equation of the line perpendicular to \(x + 8 = 0\) through \((-9, 3)\) is \(y = 3\).

Let me know if you need more details or clarifications!

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Here are five related questions to expand on this:
1. How do you find the slope of a vertical line?
2. What is the general form of the equation of a vertical line?
3. How would you write the equation of a line parallel to a given horizontal line?
4. What is the slope of a line perpendicular to a horizontal line?
5. How can we find the distance between two parallel lines?

**Tip**: Vertical lines have undefined slopes, while horizontal lines have a slope of zero.

Given point (-9, 3) and the line equation x + 8 = 0, (a) find the equation of the line parallel to the given line and passing through the point, (b) find the equation of the line perpendicular to the given line and passing through the point.

Solve a problem where you find the equations of lines parallel and perpendicular to the given line x + 8 = 0, passing through the point (-9, 3). This involves using concepts related to vertical and horizontal lines.

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