To derive the correct equation for a line through the origin based on the options provided, we can analyze each option.
A. \( y = 3x \)
B. \( y = \frac{3}{3}x \) (which simplifies to \( y = 1x \) or \( y = x \))
C. \( y = x \)
D. \( y = 8x \)
Since we are looking for equations of the form \( y = mx \) where \( m \) is the slope:
- Option A has a slope of 3.
- Option B simplifies to \( y = x \), which has a slope of 1.
- Option C is already in the form \( y = x \), which also has a slope of 1.
- Option D has a slope of 8.
All these equations represent lines through the origin, but the numerical coefficient, or slope \( m \), will determine the steepness of the line.
Thus, to choose the correct simplified form:
- If \( m = 3 \), then \( y = 3x \).
- If \( m = 8 \), then \( y = 8x \).
- If \( m = 1 \), as both options B and C show, it simplifies to \( y = x \).
Since options B and C yield the same equation \( y = x \) after simplification, it is important to note that \( B \) is a correct representation of \( y = x\) but not in its simplest, most recognizable form.
In conclusion, if asking for a line through the origin in its simplest form, you would typically use \( y = x \).
If you wanted to select from the options given:
- Most simplified form: Option C: \( y = x \).