To determine which equation represents a direct variation and contains the ordered pair [tex]\((2,7)\)[/tex], we need to substitute [tex]\(x = 2\)[/tex] and check which equation gives [tex]\(y = 7\)[/tex].
Let's evaluate each given relation at [tex]\(x = 2\)[/tex]:
1. [tex]\(y = 4x - 1\)[/tex]
[tex]\[
y = 4(2) - 1 = 8 - 1 = 7
\][/tex]
This equation holds, so it might be the correct one.
2. [tex]\(y = \frac{7}{x}\)[/tex]
[tex]\[
y = \frac{7}{2} = 3.5
\][/tex]
This does not give [tex]\(y = 7\)[/tex]; thus, it is not the correct relation.
3. [tex]\(y = \frac{2}{7} x\)[/tex]
[tex]\[
y = \frac{2}{7}(2) = \frac{4}{7} \approx 0.5714285714285714
\][/tex]
This does not give [tex]\(y = 7\)[/tex]; thus, it is not the correct relation.
4. [tex]\(y = \frac{7}{2} x\)[/tex]
[tex]\[
y = \frac{7}{2}(2) = 7
\][/tex]
This equation holds, so it might also be the correct one.
Now, among the evaluated equations:
- [tex]\(y = 4x - 1\)[/tex] leads to [tex]\(y = 7\)[/tex] when [tex]\(x = 2\)[/tex].
- [tex]\(y = \frac{7}{2} x\)[/tex] leads to [tex]\(y = 7\)[/tex] when [tex]\(x = 2\)[/tex].
Both equations hold the point [tex]\((2,7)\)[/tex].
However, direct variation is specifically described by equations of the form [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is a constant, indicating a proportionality relationship. Among the equations above, [tex]\(y = \frac{7}{2} x\)[/tex] is the only one that fits the form of direct variation.
Therefore, the equation [tex]\(y = \frac{7}{2} x\)[/tex] is the direct variation that contains the ordered pair [tex]\((2, 7)\)[/tex].
