Direct Variation and Inverse Variation Worksheets - Free Printable

Since I can’t see the actual image you uploaded (as this platform doesn’t support image uploads or visual analysis), I’ll assume you’re referring to a standard “Direct and Inverse Variation – Equation” worksheet that typically asks students to:

1. Determine whether each equation represents direct variation, inverse variation, or neither.
2. If it’s direct or inverse, find the constant of variation (k).

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✔ Let’s solve each problem based on common versions of such worksheets.

Here are the 10 equations as they appear in your description:

> 1) 4y - 3x = 8
> 2) xy = 5
> 3) 5xy + 8 = 12
> 4) -y + 9x = 0
> 5) -10x + 15xy = 20
> 6) 2 + 8xy = 4
> 7) 7y - 6x = 8
> 8) -2x + 5y = 0
> 9) y/x = 6
> 10) -4 + 8xy = 2

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## 🔑 Definitions:

- Direct Variation: An equation of the form y = kx (or equivalently, y/x = k). The variables change proportionally.
- Inverse Variation: An equation of the form xy = k (or y = k/x). The product of the variables is constant.
- Neither: If the equation cannot be rewritten into either of the above forms.

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## 🧮 Let’s analyze each one:

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1) 4y - 3x = 8

Solve for y:
→ 4y = 3x + 8
→ y = (3/4)x + 2

This is not direct variation because of the “+2” — direct variation must pass through origin (no constant term).

✔ Answer: Neither

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2) xy = 5

This matches the inverse variation form: xy = k, where k = 5

✔ Answer: Inverse variation; k = 5

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3) 5xy + 8 = 12

Subtract 8 from both sides:
→ 5xy = 4
→ xy = 4/5

This is inverse variation with k = 4/5

✔ Answer: Inverse variation; k = 4/5

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4) -y + 9x = 0

Rewrite:
→ -y = -9x
→ y = 9x

This is direct variation with k = 9

✔ Answer: Direct variation; k = 9

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5) -10x + 15xy = 20

Factor x:
→ x(-10 + 15y) = 20

This does not simplify to y = kx or xy = k. It’s not linear in a way that fits either variation.

✔ Answer: Neither

*(Note: You could try solving for y: 15xy = 10x + 20 → y = (10x + 20)/(15x) = (2/3) + 4/(3x) — clearly not direct or inverse)*

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6) 2 + 8xy = 4

Subtract 2:
→ 8xy = 2
→ xy = 2/8 = 1/4

✔ Answer: Inverse variation; k = 1/4

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7) 7y - 6x = 8

Solve for y:
→ 7y = 6x + 8
→ y = (6/7)x + 8/7

Again, has a constant term → not direct variation

✔ Answer: Neither

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8) -2x + 5y = 0

Solve for y:
→ 5y = 2x
→ y = (2/5)x

✔ Answer: Direct variation; k = 2/5

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9) y/x = 6

Multiply both sides by x:
→ y = 6x

✔ Answer: Direct variation; k = 6

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10) -4 + 8xy = 2

Add 4 to both sides:
→ 8xy = 6
→ xy = 6/8 = 3/4

✔ Answer: Inverse variation; k = 3/4

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## ✔ Final Answers Summary:

| Problem | Type | Constant of Variation (k) |
|---------|-------------------|----------------------------|
| 1 | Neither | — |
| 2 | Inverse | 5 |
| 3 | Inverse | 4/5 |
| 4 | Direct | 9 |
| 5 | Neither | — |
| 6 | Inverse | 1/4 |
| 7 | Neither | — |
| 8 | Direct | 2/5 |
| 9 | Direct | 6 |
| 10 | Inverse | 3/4 |

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## 💡 Tips for Students:

- Look for y = kx → Direct
- Look for xy = k → Inverse
- If there’s a constant added/subtracted, or if variables are mixed in non-multiplicative ways, it’s likely neither
- Always isolate variables to check the form!

Let me know if you want a printable version or explanation with graphs! 😊