To find the equation of the line that contains the points [tex]\((3,1)\)[/tex], [tex]\((9,3)\)[/tex], and [tex]\((27,9)\)[/tex], we will determine if the points satisfy a specific linear relationship.
### Step-by-Step Solution:
1. Identify the pattern among the points:
- Given points: [tex]\((3,1)\)[/tex], [tex]\((9,3)\)[/tex], and [tex]\((27,9)\)[/tex].
2. Calculate the ratios of [tex]\( \frac{y}{x} \)[/tex] for each point:
- For the point [tex]\((3, 1)\)[/tex], the ratio is [tex]\( \frac{1}{3} \)[/tex].
- For the point [tex]\((9, 3)\)[/tex], the ratio is [tex]\( \frac{3}{9} = \frac{1}{3} \)[/tex].
- For the point [tex]\((27, 9)\)[/tex], the ratio is [tex]\( \frac{9}{27} = \frac{1}{3} \)[/tex].
3. Check if all ratios are the same:
- We notice that [tex]\( \frac{1}{3} \)[/tex] is consistent for all the points. This suggests a linear relationship where the slope of the line [tex]\(m\)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
4. Form the equation of the line:
- The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex]. Given the slope ([tex]\(m\)[/tex]) is [tex]\( \frac{1}{3} \)[/tex] and recognizing that points (3,1), (9,3), and (27,9) suggest the line passes through the origin (since for [tex]\((x, y)\)[/tex], [tex]\( y \)[/tex] is largely proportional to [tex]\( x \)[/tex] with no constant term added), the intercept [tex]\(b\)[/tex] must be 0.
Thus, the equation of the line can be written as:
[tex]\[
y = \frac{1}{3} x
\][/tex]
### Conclusion:
The correct equation for the line containing the given points is:
[tex]\[
\boxed{y = \frac{1}{3} x}
\][/tex]
Thus, the correct choice is:
C. [tex]\( y = \frac{1}{3} x \)[/tex]
