To determine which relation is a direct variation that contains the ordered pair [tex]\((2,7)\)[/tex], we analyze each given equation:
1. First Relation: [tex]\( y = 4x - 1 \)[/tex]
Substitute the given point [tex]\((2, 7)\)[/tex]:
[tex]\[
y = 4(2) - 1 = 8 - 1 = 7
\][/tex]
Although the point [tex]\((2,7)\)[/tex] satisfies this equation, a direct variation is represented as [tex]\(y = kx\)[/tex] where [tex]\(k\)[/tex] is a constant. The term [tex]\(-1\)[/tex] means it is not a direct variation because the constant adjustment violates the form of direct variation.
2. Second Relation: [tex]\( y = \frac{7}{x} \)[/tex]
Substitute the given point [tex]\((2, 7)\)[/tex]:
[tex]\[
y = \frac{7}{2}
\][/tex]
For [tex]\((2,7)\)[/tex]:
[tex]\[
7 \neq \frac{7}{2}
\][/tex]
Therefore, the second relation is neither satisfied by the point [tex]\((2, 7)\)[/tex] nor is it in the form [tex]\(y = kx\)[/tex]. Hence, it is not a direct variation.
3. Third Relation: [tex]\( y = \frac{2}{7}x \)[/tex]
Substitute the given point [tex]\((2, 7)\)[/tex]:
[tex]\[
y = \frac{2}{7} \times 2 = \frac{4}{7}
\][/tex]
For [tex]\((2,7)\)[/tex]:
[tex]\[
7 \neq \frac{4}{7}
\][/tex]
Hence, the third relation is not a direct variation as it does not satisfy the point [tex]\((2, 7)\)[/tex] and is not in the correct form [tex]\(y = kx\)[/tex] for [tex]\(k\)[/tex] being a valid constant causing [tex]\(y\)[/tex] to equal to [tex]\(7\)[/tex].
4. Fourth Relation: [tex]\( y = \frac{7}{2}x \)[/tex]
Substitute the given point [tex]\((2, 7)\)[/tex]:
[tex]\[
y = \frac{7}{2} \times 2 = 7
\][/tex]
Here, the equation simplifies correctly to [tex]\(7\)[/tex]. Furthermore, it is in the correct format of a direct variation [tex]\(y = kx\)[/tex], specifically [tex]\(k = \frac{7}{2}\)[/tex].
Given these results, the fourth relation [tex]\( y = \frac{7}{2}x \)[/tex] is a direct variation that includes the ordered pair [tex]\((2, 7)\)[/tex].
Thus, the correct relation is:
[tex]\[
y = \frac{7}{2}x
\][/tex]
