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### Part 1: Parallel and Perpendicular
The graph shows a red line, and you need to answer the following questions:

#### 1. Write an equation for a line parallel to the one shown in the graph.
   - A line parallel to another will have the same slope but may have a different y-intercept.
   
   Let's first find the slope of the given line. To do that, we can pick two points on the line. From the graph:
   - The line passes through points approximately at (0, 2) and (6, 0).

   Using these two points, we can calculate the slope (m):
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 2}{6 - 0} = \frac{-2}{6} = -\frac{1}{3}
   \]
   
   The slope of the original line is \(-\frac{1}{3}\). A parallel line will have the same slope.

   Thus, the general equation for a line parallel to this one would be in the form:
   \[
   y = -\frac{1}{3}x + b
   \]
   To find the specific equation, you would need a point that the parallel line passes through. Since no point is given, the equation remains general for now.

#### 2. Write an equation for a line perpendicular to the one shown in the graph and passing through (2, 5).
   - A line perpendicular to the given line will have a slope that is the negative reciprocal of \(-\frac{1}{3}\), which is 3.

   The slope of the perpendicular line is 3, and it passes through the point (2, 5). We can use the point-slope form of the equation:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \(m = 3\), \(x_1 = 2\), and \(y_1 = 5\):
   \[
   y - 5 = 3(x - 2)
   \]
   Simplifying:
   \[
   y - 5 = 3x - 6 \quad \Rightarrow \quad y = 3x - 1
   \]
   So, the equation of the line perpendicular to the given one and passing through (2, 5) is:
   \[
   y = 3x - 1
   \]

Would you like more details on any part, or do you have any questions so far?

Here are 5 related questions you might find helpful:
1. What is the relationship between the slopes of parallel lines?
2. How do you determine the equation of a line if given two points?
3. What is the significance of the y-intercept in a linear equation?
4. Can two different lines have the same slope and still not be parallel?
5. How would you find the equation of a line that passes through a given point and is neither parallel nor perpendicular to the given line?

**Tip:** Always double-check the coordinates you use when calculating the slope to avoid errors.

Using the graph above, write an equation that is parallel to the line in the graph. Show your work and explain how you came up with your equation. Write an equation that is perpendicular to the line in the graph and goes through the point (2, 5). Show your work and explain how you came up with your equation.

Learn how to find equations of lines that are parallel and perpendicular to a given line using graphs. This guide includes step-by-step instructions on calculating slopes and using the point-slope form, suitable for Grades 8-10 students.

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