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The problem presented involves the line given by the equation \( y = \frac{9}{8}x + 8 \). The task is to find the equations of two lines:

1. A line parallel to the given line that passes through the point \( P(8, 5) \).
2. A line perpendicular to the given line that passes through the same point \( P(8, 5) \).

### 1. Equation of the Parallel Line:
The equation of any line parallel to the given line will have the same slope. The slope \( m \) of the given line is \( \frac{9}{8} \).

So, the equation of the line parallel to this one that passes through the point \( P(8, 5) \) can be written in point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substituting the point \( P(8, 5) \) and the slope \( m = \frac{9}{8} \):
\[
y - 5 = \frac{9}{8}(x - 8)
\]
Simplifying, we get:
\[
y - 5 = \frac{9}{8}x - 9
\]
\[
y = \frac{9}{8}x - 9 + 5
\]
\[
y = \frac{9}{8}x - 4
\]
So, the equation of the parallel line is:
\[
y = \frac{9}{8}x - 4
\]

### 2. Equation of the Perpendicular Line:
The slope of the line perpendicular to the given line will be the negative reciprocal of \( \frac{9}{8} \). The negative reciprocal is \( -\frac{8}{9} \).

Using the point-slope form again for the point \( P(8, 5) \):
\[
y - 5 = -\frac{8}{9}(x - 8)
\]
Simplifying, we get:
\[
y - 5 = -\frac{8}{9}x + \frac{64}{9}
\]
\[
y = -\frac{8}{9}x + \frac{64}{9} + 5
\]
Since \( 5 = \frac{45}{9} \), we have:
\[
y = -\frac{8}{9}x + \frac{64}{9} + \frac{45}{9}
\]
\[
y = -\frac{8}{9}x + \frac{109}{9}
\]
So, the equation of the perpendicular line is:
\[
y = -\frac{8}{9}x + \frac{109}{9}
\]

Would you like further details or have any other questions?

### Related Questions:
1. How do you derive the point-slope form of a line?
2. What are the steps to find the slope of a line given its equation?
3. How do you find the equation of a line given two points?
4. Can you determine the distance between two parallel lines?
5. What is the significance of the negative reciprocal in finding perpendicular lines?

### Tip:
Always verify the slope when determining whether lines are parallel or perpendicular.

Explore how to find equations of parallel and perpendicular lines to y = 9/8x + 8 passing through point P(8, 5). This problem covers concepts of lines, slopes, and point-slope form. Suitable for high school students.

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