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Let's analyze and solve this problem step by step.

The given line is:

\[
y = -\frac{4}{7}x - 3
\]

### 1. **Equation of the Parallel Line**

For a line to be parallel to the given one, it must have the **same slope**. The slope of the given line is \( -\frac{4}{7} \). The parallel line will pass through the point \( (8, -3) \).

We use the **point-slope form** of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

Where:
- \( m \) is the slope,
- \( (x_1, y_1) \) is the point the line passes through.

Substitute \( m = -\frac{4}{7} \), \( x_1 = 8 \), and \( y_1 = -3 \):

\[
y - (-3) = -\frac{4}{7}(x - 8)
\]

Simplifying:

\[
y + 3 = -\frac{4}{7}(x - 8)
\]

Distribute the slope:

\[
y + 3 = -\frac{4}{7}x + \frac{32}{7}
\]

Subtract 3 from both sides to isolate \( y \):

\[
y = -\frac{4}{7}x + \frac{32}{7} - 3
\]

Convert 3 to a fraction:

\[
y = -\frac{4}{7}x + \frac{32}{7} - \frac{21}{7}
\]

Simplify:

\[
y = -\frac{4}{7}x + \frac{11}{7}
\]

So, the equation of the parallel line is:

\[
y = -\frac{4}{7}x + \frac{11}{7}
\]

### 2. **Equation of the Perpendicular Line**

For the perpendicular line, we need the **negative reciprocal** of the slope \( -\frac{4}{7} \), which is \( \frac{7}{4} \).

Again, use the point-slope form. This line also passes through the point \( (8, -3) \), so we substitute \( m = \frac{7}{4} \), \( x_1 = 8 \), and \( y_1 = -3 \):

\[
y - (-3) = \frac{7}{4}(x - 8)
\]

Simplifying:

\[
y + 3 = \frac{7}{4}(x - 8)
\]

Distribute the slope:

\[
y + 3 = \frac{7}{4}x - 14
\]

Subtract 3 from both sides:

\[
y = \frac{7}{4}x - 14 - 3
\]

Simplify:

\[
y = \frac{7}{4}x - 17
\]

So, the equation of the perpendicular line is:

\[
y = \frac{7}{4}x - 17
\]

### Final Answers:

1. **Equation of the parallel line**: \( y = -\frac{4}{7}x + \frac{11}{7} \)
2. **Equation of the perpendicular line**: \( y = \frac{7}{4}x - 17 \)

Let me know if you'd like further details or have any questions!

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**Related Questions:**
1. What is the formula for finding the slope of a line given two points?
2. How do you determine the slope of a line from its equation in general form?
3. What is the relationship between the slopes of perpendicular lines?
4. Can you derive the point-slope form of a line from its slope-intercept form?
5. How do you convert a line equation from point-slope form to slope-intercept form?

**Tip**: Always remember, parallel lines have the same slope, and perpendicular lines have slopes that are negative reciprocals of each other.

Consider the line y = -4/7x - 3. Find the equation of the line that is parallel to this line and passes through the point (8, -3). Find the equation of the line that is perpendicular to this line and passes through the point (8, -3).

Solve the problem of finding the equations of lines parallel and perpendicular to y = -4/7x - 3, passing through the point (8, -3). This solution involves linear equations, point-slope form, and negative reciprocals, suitable for students in Grades 9-12.

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