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Let's solve the problem step by step:

### 1. **Given Line Equation**:
The given equation of the line is:
\[
y = -\frac{4}{7}x - 1
\]
The slope of this line is the coefficient of \(x\), which is:
\[
m_1 = -\frac{4}{7}
\]
### 2. **Parallel Line**:
A line that is parallel to the given line will have the same slope. Thus, the slope \(m_2\) of the parallel line is also:
\[
m_2 = -\frac{4}{7}
\]
We are given the point \((4, -3)\) through which this parallel line passes. To find the equation of the parallel line, we use the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
Substituting \(m = -\frac{4}{7}\), \(x_1 = 4\), and \(y_1 = -3\):
\[
y - (-3) = -\frac{4}{7}(x - 4)
\]
Simplifying this:
\[
y + 3 = -\frac{4}{7}(x - 4)
\]
\[
y + 3 = -\frac{4}{7}x + \frac{16}{7}
\]
\[
y = -\frac{4}{7}x + \frac{16}{7} - 3
\]
\[
y = -\frac{4}{7}x + \frac{16}{7} - \frac{21}{7}
\]
\[
y = -\frac{4}{7}x - \frac{5}{7}
\]
Thus, the equation of the parallel line is:
\[
y = -\frac{4}{7}x - \frac{5}{7}
\]

### 3. **Perpendicular Line**:
For the perpendicular line, the slope will be the negative reciprocal of the original slope. So, if the slope of the original line is \(m_1 = -\frac{4}{7}\), the slope of the perpendicular line will be:
\[
m_3 = \frac{7}{4}
\]
Again, using the point-slope form and the point \((4, -3)\), we have:
\[
y - (-3) = \frac{7}{4}(x - 4)
\]
Simplifying this:
\[
y + 3 = \frac{7}{4}(x - 4)
\]
\[
y + 3 = \frac{7}{4}x - 7
\]
\[
y = \frac{7}{4}x - 7 - 3
\]
\[
y = \frac{7}{4}x - 10
\]
Thus, the equation of the perpendicular line is:
\[
y = \frac{7}{4}x - 10
\]

### Final Answers:
- **Equation of the parallel line**: \( y = -\frac{4}{7}x - \frac{5}{7} \)
- **Equation of the perpendicular line**: \( y = \frac{7}{4}x - 10 \)

Let me know if you would like more details or have any questions.

#### Related questions:
1. How do you calculate the slope of a line from its equation?
2. What is the significance of a slope being parallel to another line?
3. What is the general method for finding the negative reciprocal of a slope?
4. Can you explain how the point-slope form of a line works?
5. How would the process differ if the point did not lie on the line?

**Tip**: When dealing with lines, always verify the slope first. Parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals.

Find the equation of the line that is parallel to y = -4/7x - 1 and passes through the point (4, -3). Also, find the equation of the line that is perpendicular to this line and passes through the same point.

Learn how to find the equation of a line parallel and perpendicular to y = -4/7x - 1 that passes through the point (4, -3). This detailed solution covers key algebraic concepts, including slope relationships for parallel and perpendicular lines.

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