A proportional relationship means that the ratio \( \frac{y}{x} \) remains constant for all pairs of \( (x, y) \).
Let’s analyze each table:
Table A:
x | y
----------
2 | 9 → \\(\frac{9}{2} = 4.5\\)
4 | 18 → \\(\frac{18}{4} = 4.5\\)
6 | 27 → \\(\frac{27}{6} = 4.5\\)
8 | 36 → \\(\frac{36}{8} = 4.5\\)
10 | 45 → \\(\frac{45}{10} = 4.5\\)
The ratios are all equal (\(4.5\)), so Table A shows a proportional relationship.
Table B:
x | y
----------
2 | 2 → \\(\frac{2}{2} = 1\\)
4 | 8 → \\(\frac{8}{4} = 2\\)
6 | 18 → \\(\frac{18}{6} = 3\\)
8 | 32 → \\(\frac{32}{8} = 4\\)
10 | 50 → \\(\frac{50}{10} = 5\\)
The ratios are not equal, so Table B does not show a proportional relationship.
Table C:
x | y
----------
1 | 4 → \\(\frac{4}{1} = 4\\)
2 | 7 → \\(\frac{7}{2} = 3.5\\)
3 | 10 → \\(\frac{10}{3} \approx 3.33\\)
4 | 13 → \\(\frac{13}{4} = 3.25\\)
5 | 16 → \\(\frac{16}{5} = 3.2\\)
The ratios are not equal, so Table C does not show a proportional relationship.
Table D:
x | y
----------
1 | 5 → \\(\frac{5}{1} = 5\\)
2 | 6 → \\(\frac{6}{2} = 3\\)
3 | 7 → \\(\frac{7}{3} \approx 2.33\\)
4 | 8 → \\(\frac{8}{4} = 2\\)
5 | 9 → \\(\frac{9}{5} = 1.8\\)
The ratios are not equal, so Table D does not show a proportional relationship.
Conclusion: Only Table A shows a proportional relationship between \( x \) and \( y \).