To solve this problem, we can determine the equation of a line using the given intercepts:

• The x-intercept is 8, meaning the point $(8, 0)$ lies on the line.
• The y-intercept is 4, meaning the point $(0, 4)$ lies on the line.

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

$y = mx + b$

Where:

• $m$ is the slope
• $b$ is the y-intercept (which is already given as 4)

Step 1: Calculate the slope

The slope $m$ is calculated using the formula:

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

Using the points $(8, 0)$ and $(0, 4)$:

$m = \frac{4 - 0}{0 - 8} = \frac{4}{-8} = -\frac{1}{2}$

Step 2: Write the equation

Now that we know the slope $m = -\frac{1}{2}$ and the y-intercept is 4, we can write the equation as:

$y = -\frac{1}{2}x + 4$

Step 3: Match with the given options

This matches Option C, which is:

$y = -\frac{1}{2}x + 4$

Thus, the correct answer is C.

Would you like further explanation on any part? Here are five related questions to deepen your understanding:

1. How do you find the slope of a line given two points?
2. Can you derive the equation of a line if only one point and the slope are given?
3. What is the significance of the y-intercept in the slope-intercept form?
4. How can we find the x-intercept of a line if only the equation is given?
5. How does the sign of the slope affect the graph of the line?

Tip: When finding the slope, always ensure to subtract the coordinates in the same order (e.g., $y_2 - y_1$ and $x_2 - x_1$).