Question

A proportional linear relationship means that there is a constant ratio between the \(x\) and \(y\) values, which can be determined by checking if the ratio \(\frac{y}{x}\) is the same for all pairs.

Let's analyze each table:

1. **First Table:**
- \( (3, 5) \) → \( \frac{5}{3} \)
- \( (4, 7) \) → \( \frac{7}{4} \)
- \( (5, 9) \) → \( \frac{9}{5} \)
- \( (6, 11) \) → \( \frac{11}{6} \)
- These ratios are not constant.

2. **Second Table:**
- \( (1, 5) \) → \( \frac{5}{1} = 5 \)
- \( (2, 6) \) → \( \frac{6}{2} = 3 \)
- \( (3, 7) \) → \( \frac{7}{3} \)
- \( (4, 8) \) → \( \frac{8}{4} = 2 \)
- These ratios are not constant.

3. **Third Table:**
- \( (2, 3) \) → \( \frac{3}{2} \)
- \( (4, 6) \) → \( \frac{6}{4} = \frac{3}{2} \)
- \( (10, 15) \) → \( \frac{15}{10} = \frac{3}{2} \)
- \( (14, 21) \) → \( \frac{21}{14} = \frac{3}{2} \)
- All ratios are constant (equal to \(\frac{3}{2}\)).

4. **Fourth Table:**
- \( (2, 4) \) → \( \frac{4}{2} = 2 \)
- \( (3, 9) \) → \( \frac{9}{3} = 3 \)
- \( (4, 16) \) → \( \frac{16}{4} = 4 \)
- \( (5, 25) \) → \( \frac{25}{5} = 5 \)
- These ratios are not constant.

Based on this analysis, **the third table represents a proportional linear relationship.**