To determine which tables represent a proportional relationship between \( x \) and \( y \), we need to check if the ratio \( \frac{y}{x} \) is constant for all values of \( x \) (excluding \( x = 0 \) where \( y \) should be \( 0 \)).
Let's analyze each table:
Table A:
- \( x = 1, y = 6 \Rightarrow \frac{y}{x} = \frac{6}{1} = 6 \)
- \( x = 2, y = 9 \Rightarrow \frac{y}{x} = \frac{9}{2} = 4.5 \)
- \( x = 3, y = 12 \Rightarrow \frac{y}{x} = \frac{12}{3} = 4 \)
Conclusion: The ratios are not constant. Not proportional.
Table B:
- \( x = 2, y = 12 \Rightarrow \frac{y}{x} = \frac{12}{2} = 6 \)
- \( x = 3, y = 18 \Rightarrow \frac{y}{x} = \frac{18}{3} = 6 \)
- \( x = 4, y = 24 \Rightarrow \frac{y}{x} = \frac{24}{4} = 6 \)
- \( x = 5, y = 30 \Rightarrow \frac{y}{x} = \frac{30}{5} = 6 \)
Conclusion: The ratios are constant at 6. This is proportional.
Table C:
- \( x = 2, y = 0 \Rightarrow \frac{y}{x} = \frac{0}{2} = 0 \)
- \( x = 4, y = 2 \Rightarrow \frac{y}{x} = \frac{2}{4} = 0.5 \)
- \( x = 6, y = 4 \Rightarrow \frac{y}{x} = \frac{4}{6} \approx 0.67 \)
- \( x = 9, y = 6 \Rightarrow \frac{y}{x} = \frac{6}{9} \approx 0.67 \)
Conclusion: The ratios are not constant. Not proportional.
Table D:
- \( x = 1, y = 2 \Rightarrow \frac{y}{x} = \frac{2}{1} = 2 \)
- \( x = 2, y = 5 \Rightarrow \frac{y}{x} = \frac{5}{2} = 2.5 \)
- \( x = 3, y = 9 \Rightarrow \frac{y}{x} = \frac{9}{3} = 3 \)
Conclusion: The ratios are not constant. Not proportional.
Final Answer:
The only table that represents a proportional relationship between \( x \) and \( y \) is:
B