To determine whether the line through the points [tex]\((-4, 3)\)[/tex] and [tex]\((4, 3)\)[/tex] represents a direct variation, let's first understand what direct variation means in the context of lines.
1. Understanding Direct Variation:
- A direct variation can be expressed in the form [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is a constant, and the line must pass through the origin [tex]\((0,0)\)[/tex].
- If a line goes through the origin and can be written in the form [tex]\(y = kx\)[/tex], then it represents a direct variation.
2. Examining the Given Points:
- We have two points [tex]\((-4, 3)\)[/tex] and [tex]\((4, 3)\)[/tex].
- Notice that both points have the same [tex]\(y\)[/tex]-coordinate, which means the line is horizontal and has the equation [tex]\(y = 3\)[/tex].
3. Checking if the Line Passes Through the Origin:
- The origin is the point [tex]\((0,0)\)[/tex].
- For the line [tex]\(y = 3\)[/tex] to pass through the origin, substituting [tex]\(x = 0\)[/tex] into the equation must yield [tex]\(y = 0\)[/tex].
- Substituting [tex]\(x = 0\)[/tex] into [tex]\(y = 3\)[/tex] gives [tex]\(y = 3\)[/tex], which is not equal to 0. Therefore, the line does not pass through the origin.
4. Conclusion:
- Since the line does not pass through the origin, it cannot represent a direct variation.
Therefore, the correct description is:
The line does not represent a direct variation because it does not go through the origin.
