To solve the problem, we need to determine the equation of a line that:

1. Is perpendicular to the green line.
2. Passes through the purple point.

The equation of a line in slope-intercept form is given by:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Step 1: Find the slope of the green line

We can visually estimate the slope of the green line by selecting two points on the green line and using the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

From the graph:

• The green line passes through the point $(0, -10)$ and the point $(5, -5)$.
• Using these points, the slope $m_{\text{green}}$ is:

$m_{\text{green}} = \frac{-5 - (-10)}{5 - 0} = \frac{5}{5} = 1$

So, the slope of the green line is $m_{\text{green}} = 1$.

Step 2: Find the slope of the perpendicular line

If two lines are perpendicular, their slopes are negative reciprocals of each other. Therefore, the slope $m_{\text{perpendicular}}$ of the line perpendicular to the green line is:

$m_{\text{perpendicular}} = -\frac{1}{1} = -1$

Step 3: Use the point-slope form

Now, we use the purple point to find the y-intercept of the perpendicular line. The coordinates of the purple point are $(20, 0)$.

Using the slope-intercept form $y = mx + b$ with $m = -1$ and the point $(20, 0)$:

$0 = -1(20) + b$

Solving for $b$:

$0 = -20 + b \implies b = 20$

Thus, the equation of the line is:

$y = -x + 20$

Step 4: Compare with the answer choices

The correct answer is:

$y = -x + 20$

However, none of the provided options match this exactly, suggesting a potential discrepancy with the given choices or a small detail in their labeling. Based on the graph and analysis, the equation $y = -x + 20$ is correct.

Let me know if you need more clarification or if you have any further questions! Here are 5 relative questions to consider:

1. How do you find the slope of a line given two points?
2. What is the relationship between the slopes of perpendicular lines?
3. How does point-slope form help in finding the equation of a line?
4. How do you convert point-slope form to slope-intercept form?
5. What would happen if the line passed through a different point on the graph?

Tip: When working with slopes, always check for negative reciprocal relationships when dealing with perpendicular lines.