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!

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!