Let's break down the task based on the image you uploaded.

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.