https://file.aiquickdraw.com/mathbot/file/172782596544469jgh9gn.jpg

Let's first work through the problem given in the image.

### Given Line:
\[
-8x - 7y = 9
\]
We'll find:
1. The equation of the line perpendicular to this given line and passing through the point \( (5, -3) \).
2. The equation of the line parallel to this given line and passing through the point \( (5, -3) \).

### Step 1: Write the equation of the given line in slope-intercept form
The slope-intercept form is \( y = mx + b \), where \( m \) is the slope. To do that, solve for \( y \):
\[
-8x - 7y = 9 \quad \text{(move \( -8x \) to the other side)}
\]
\[
-7y = 8x + 9
\]
\[
y = \frac{-8}{7}x - \frac{9}{7}
\]
So, the slope of the given line is \( m = \frac{-8}{7} \).

### Step 2: Find the equation of the line perpendicular to the given line
The slope of the line perpendicular to a given line is the negative reciprocal of the original slope. So, the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = \frac{7}{8}
\]
Now, use the point-slope form \( y - y_1 = m(x - x_1) \) with the point \( (5, -3) \):
\[
y + 3 = \frac{7}{8}(x - 5)
\]
Simplify this:
\[
y + 3 = \frac{7}{8}x - \frac{35}{8}
\]
\[
y = \frac{7}{8}x - \frac{35}{8} - 3
\]
Convert \( -3 \) to a fraction:
\[
y = \frac{7}{8}x - \frac{35}{8} - \frac{24}{8}
\]
\[
y = \frac{7}{8}x - \frac{59}{8}
\]
Thus, the equation of the perpendicular line is:
\[
y = \frac{7}{8}x - \frac{59}{8}
\]

### Step 3: Find the equation of the line parallel to the given line
The slope of the parallel line is the same as the given line, \( m = \frac{-8}{7} \). Using the point \( (5, -3) \) in the point-slope form:
\[
y + 3 = \frac{-8}{7}(x - 5)
\]
Simplify this:
\[
y + 3 = \frac{-8}{7}x + \frac{40}{7}
\]
\[
y = \frac{-8}{7}x + \frac{40}{7} - 3
\]
Convert \( -3 \) to a fraction:
\[
y = \frac{-8}{7}x + \frac{40}{7} - \frac{21}{7}
\]
\[
y = \frac{-8}{7}x + \frac{19}{7}
\]
Thus, the equation of the parallel line is:
\[
y = \frac{-8}{7}x + \frac{19}{7}
\]

### Final Answers:
- Equation of the perpendicular line: \( y = \frac{7}{8}x - \frac{59}{8} \)
- Equation of the parallel line: \( y = \frac{-8}{7}x + \frac{19}{7} \)

Would you like further details or have any questions?

Here are five related questions you might consider:
1. How do we find the distance between two parallel lines?
2. What happens to the slopes of perpendicular lines in 3D space?
3. How can we determine the intersection point of two lines?
4. How does changing a point on a line affect its equation?
5. Can two lines with different slopes ever be parallel?

**Tip:** Always convert fractions early when possible to avoid working with large numbers!

Find the equation of the line that is perpendicular to -8x - 7y = 9 and passes through the point (5, -3). Also, find the equation of the line that is parallel to the same line and passes through the point (5, -3).

Learn how to find the equations of perpendicular and parallel lines to the equation -8x - 7y = 9, passing through the point (5, -3). This solution covers key algebra concepts, including slopes, point-slope form, and properties of perpendicular and parallel lines. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation