To derive the equation of a line through the origin (which has the form \( y = mx \), where \( m \) is the slope), we need to calculate the slope using the points given: \( (8, 8) \) and \( (3, 3) \).
The slope \( m \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (8, 8) \) as \( (x_1, y_1) \) and \( (3, 3) \) as \( (x_2, y_2) \):
\[ m = \frac{3 - 8}{3 - 8} = \frac{-5}{-5} = 1 \]
Therefore, the slope \( m \) is 1, which gives us the equation:
\[ y = 1x \quad \text{or} \quad y = x \]
Since the question asks us to simplify the numerical coefficient if needed, we see that \( 1x \) simplifies to just \( x \).
Now, looking at the provided options:
A. \( y = 8x \)
B. \( y = \frac{3}{3}x \) (which simplifies to \( y = x \))
C. \( y = x \)
D. \( y = 3x \)
The correct answer that matches our derived equation \( y = x \) is:
C. \( y = x \)