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The problem in the image asks to find two equations:

1. The equation of a line that is parallel to the given line \( y = \frac{6}{7}x + 1 \) and passes through the point \( (7, 2) \).
2. The equation of a line that is perpendicular to the given line and passes through the same point \( (7, 2) \).

### Let's break this down:

#### 1. **Parallel Line:**
Parallel lines have the same slope. The slope of the given line \( y = \frac{6}{7}x + 1 \) is \( \frac{6}{7} \). So, the parallel line will also have a slope of \( \frac{6}{7} \).

Using the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point it passes through. Substituting the slope \( m = \frac{6}{7} \) and the point \( (7, 2) \):
\[
y - 2 = \frac{6}{7}(x - 7)
\]
Simplifying:
\[
y - 2 = \frac{6}{7}x - 6
\]
\[
y = \frac{6}{7}x - 6 + 2
\]
\[
y = \frac{6}{7}x - 4
\]
So, the equation of the parallel line is:
\[
y = \frac{6}{7}x - 4
\]

#### 2. **Perpendicular Line:**
Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the given line is \( \frac{6}{7} \), so the slope of the perpendicular line will be \( -\frac{7}{6} \).

Using the point-slope form again with \( m = -\frac{7}{6} \) and \( (7, 2) \):
\[
y - 2 = -\frac{7}{6}(x - 7)
\]
Simplifying:
\[
y - 2 = -\frac{7}{6}x + \frac{49}{6}
\]
\[
y = -\frac{7}{6}x + \frac{49}{6} + 2
\]
\[
y = -\frac{7}{6}x + \frac{49}{6} + \frac{12}{6}
\]
\[
y = -\frac{7}{6}x + \frac{61}{6}
\]
So, the equation of the perpendicular line is:
\[
y = -\frac{7}{6}x + \frac{61}{6}
\]

### Summary:
- Equation of the parallel line: \( y = \frac{6}{7}x - 4 \)
- Equation of the perpendicular line: \( y = -\frac{7}{6}x + \frac{61}{6} \)

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

Here are 5 related questions you can explore:
1. How do you derive the equation of a line from two points?
2. What is the significance of the point-slope form in geometry?
3. How do you verify if two lines are perpendicular using slopes?
4. What is the difference between parallel and perpendicular slopes?
5. How do you convert between different forms of linear equations (e.g., point-slope to slope-intercept)?

**Tip**: Remember that parallel lines have equal slopes, while perpendicular lines have slopes that multiply to -1.

Find the equation of a line that is parallel to the given line y = 6/7x + 1 and passes through the point (7, 2). Also, find the equation of a line that is perpendicular to this line and passes through the same point (7, 2).

Solve for the equations of parallel and perpendicular lines to y = 6/7x + 1 that pass through the point (7, 2). Explore linear equation concepts, point-slope form, and the relationship between slopes of parallel and perpendicular lines.

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