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We are given the equation of the line \( y = -x - 9 \), and we need to find the following:

1. The equation of the line parallel to this given line that passes through the point \( (1, 8) \).
2. The equation of the line perpendicular to this given line that passes through the point \( (1, 8) \).

### 1. Parallel Line:
The slope of the given line is \( m = -1 \) (from \( y = mx + b \), where \( m \) is the slope). For a line to be parallel, it must have the same slope. Therefore, the equation of the parallel line will have the same slope \( m = -1 \), but a different y-intercept.

The equation of the line can be written in point-slope form:

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

Where \( (x_1, y_1) = (1, 8) \) and \( m = -1 \). Plugging in the values:

\[
y - 8 = -1(x - 1)
\]
\[
y - 8 = -x + 1
\]
\[
y = -x + 9
\]

So, the equation of the line parallel to \( y = -x - 9 \) that passes through \( (1, 8) \) is:
\[
y = -x + 9
\]

### 2. Perpendicular Line:
For a line to be perpendicular, its slope is the negative reciprocal of the original line's slope. Since the original slope is \( m = -1 \), the slope of the perpendicular line will be \( m = 1 \).

Again, we use the point-slope form, where \( (x_1, y_1) = (1, 8) \) and \( m = 1 \):

\[
y - 8 = 1(x - 1)
\]
\[
y - 8 = x - 1
\]
\[
y = x + 7
\]

So, the equation of the line perpendicular to \( y = -x - 9 \) that passes through \( (1, 8) \) is:
\[
y = x + 7
\]

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

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Here are 5 related questions to deepen understanding:
1. How do you find the slope of a line from its equation?
2. What is the general form of a line's equation, and how do you convert between forms?
3. How do you find the equation of a line when given two points?
4. What does it mean for two lines to be perpendicular in terms of slopes?
5. How does changing the y-intercept affect the graph of a line?

**Tip:** Always remember that parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals of each other!

Consider the line with the equation y = -x - 9. Give the equation of the line parallel to Line 1 which passes through (1, 8). Give the equation of the line perpendicular to Line 1 which passes through (1, 8).

Learn how to find the equation of lines parallel and perpendicular to y = -x - 9 passing through the point (1, 8). Explore the point-slope form of a line and understand the properties of parallel and perpendicular slopes. Suitable for Grades 8-10.

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