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To determine if the values in each table are proportional, we need to check whether the ratio of Y to X (i.e., Y/X) is the same constant for all pairs of values in the table.

> 💡 Proportionality Rule:
> Two variables X and Y are proportional if Y = k·X for some constant k.
> This means Y/X = k for every row — the ratio must be constant.

We’ll go through each table one by one.

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Table 1:

| X | Y |
|----|----|
| -4 | -5 |
| -3 | -6 |
| -2 | -7 |
| -1 | -8 |

Compute Y/X:
- -5 / -4 = 1.25
- -6 / -3 = 2
- -7 / -2 = 3.5
- -8 / -1 = 8

✘ Ratios are NOT constant → Not Proportional

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Table 2:

| X | Y |
|---|----|
| 6 | -2 |
| 7 | -1 |
| 8 | 0 |
| 9 | 1 |

Compute Y/X:
- -2 / 6 ≈ -0.333
- -1 / 7 ≈ -0.143
- 0 / 8 = 0
- 1 / 9 ≈ 0.111

✘ Ratios are NOT constant → Not Proportional

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Table 3:

| X | Y |
|-----|----|
| -20 | -10 |
| -56 | -8 |
| -14 | -2 |
| -7 | -1 |

Compute Y/X:
- -10 / -20 = 0.5
- -8 / -56 ≈ 0.1429
- -2 / -14 ≈ 0.1429
- -1 / -7 ≈ 0.1429

Wait — first one is 0.5, others are ~0.1429 → ✘ Not constant → Not Proportional

*(Note: If you check Y = kX, for last three: -8 = k*(-56) → k=1/7; -2 = k*(-14) → k=1/7; -1 = k*(-7) → k=1/7. But first pair: -10 = k*(-20) → k=0.5 ≠ 1/7. So not proportional.)*

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Table 4:

| X | Y |
|---|---|
| 6 | 3 |
| 7 | 4 |
| 8 | 5 |
| 9 | 6 |

Y/X:
- 3/6 = 0.5
- 4/7 ≈ 0.571
- 5/8 = 0.625
- 6/9 ≈ 0.666

✘ Not constant → Not Proportional

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Table 5:

| X | Y |
|----|-----|
| 1 | 10 |
| 2 | 20 |
| 7 | 70 |
| 10 | 100 |

Y/X:
- 10/1 = 10
- 20/2 = 10
- 70/7 = 10
- 100/10 = 10

✔ All ratios = 10 → Proportional

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Table 6:

| X | Y |
|---|---|
| 2 | 2 |
| 4 | 4 |
| 8 | 8 |
| 10| 10|

Y/X:
- 2/2 = 1
- 4/4 = 1
- 8/8 = 1
- 10/10 = 1

✔ All ratios = 1 → Proportional

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Table 7:

| X | Y |
|----|-----|
| 20 | -32 |
| 15 | -24 |
| 10 | -16 |
| 5 | -8 |

Y/X:
- -32/20 = -1.6
- -24/15 = -1.6
- -16/10 = -1.6
- -8/5 = -1.6

✔ All ratios = -1.6 → Proportional

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Table 8:

| X | Y |
|----|-----|
| 70 | -10 |
| 63 | -9 |
| 35 | -5 |
| 21 | -3 |

Y/X:
- -10/70 ≈ -0.1429
- -9/63 ≈ -0.1429
- -5/35 ≈ -0.1429
- -3/21 ≈ -0.1429

✔ All ratios = -1/7 → Proportional

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Table 9:

| X | Y |
|----|----|
| 2 | 7 |
| 6 | 21 |
| 18 | 63 |
| 20 | 70 |

Y/X:
- 7/2 = 3.5
- 21/6 = 3.5
- 63/18 = 3.5
- 70/20 = 3.5

✔ All ratios = 3.5 → Proportional

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Table 10:

| X | Y |
|-----|-----|
| -12 | -32 |
| -9 | -24 |
| -6 | -16 |
| -3 | -8 |

Y/X:
- -32/-12 ≈ 2.666...
- -24/-9 ≈ 2.666...
- -16/-6 ≈ 2.666...
- -8/-3 ≈ 2.666...

✔ All ratios = 8/3 → Proportional

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Table 11:

| X | Y |
|----|---|
| 9 | 3 |
| 36 | 6 |
| 64 | 8 |
| 81 | 9 |

Y/X:
- 3/9 = 0.333...
- 6/36 = 0.166...
- 8/64 = 0.125
- 9/81 = 0.111...

✘ Not constant → Not Proportional

*(Note: These look like square roots: √9=3, √36=6, √64=8, √81=9 — so Y = √X, which is NOT linear/proportional.)*

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Table 12:

| X | Y |
|---|----|
| 2 | 4 |
| 3 | 6 |
| 4 | 12 |
| 7 | 21 |

Y/X:
- 4/2 = 2
- 6/3 = 2
- 12/4 = 3
- 21/7 = 3

First two = 2, last two = 3 → ✘ Not constant → Not Proportional

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## ✔ Final Answers:

1. No
2. No
3. No
4. No
5. Yes
6. Yes
7. Yes
8. Yes
9. Yes
10. Yes
11. No
12. No

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📌 Summary Tip: Always compute Y/X for each row. If they’re all equal → proportional. If any differ → not proportional. Don’t forget negative numbers and fractions!

Let me know if you’d like a graph or formula explanation too!