Direct Variation Worksheet 2020-2024 - Fill and Sign Printable ... - Free Printable

It looks like you've uploaded a Direct Variation Worksheet with several problems related to direct variation. Since I can't view images directly, I’ll help you solve the problems based on the text you’ve provided. Let's go step by step and solve each part.

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🔹 What is Direct Variation?

A direct variation is a relationship between two variables $ x $ and $ y $ that can be written in the form:
$$
y = kx
$$
where $ k $ is the constant of variation (a constant number).

Key points:
- The graph is a straight line passing through the origin (0, 0).
- If $ y $ varies directly with $ x $, then $ \frac{y}{x} = k $ for all values.
- To find $ k $: $ k = \frac{y}{x} $

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## ✔ Problem 1: Find the constant of variation (k) or explain why it’s not direct variation

We are given equations. We need to determine if they represent direct variation ($ y = kx $), and if so, find $ k $. Otherwise, explain why not.

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1. $ 2y + 3x = 3 $

Solve for $ y $:

$$
2y = -3x + 3 \\
y = -\frac{3}{2}x + \frac{3}{2}
$$

This is not in the form $ y = kx $ because of the constant term $ \frac{3}{2} $.
➡️ Not direct variation — has a y-intercept ≠ 0.

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2. $ -6 = 10y $

Solve for $ y $:

$$
y = -\frac{6}{10} = -\frac{3}{5}
$$

So $ y = -\frac{3}{5} $, which is a horizontal line, constant value regardless of $ x $.
This is not direct variation — no dependence on $ x $, and does not pass through origin unless $ y=0 $.
➡️ Not direct variation

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3. $ -12x = 6y $

Solve for $ y $:

$$
6y = -12x \\
y = -2x
$$

This is $ y = kx $ with $ k = -2 $.
✔ Direct variation — constant of variation: $ \boxed{-2} $

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4. $ y = 8 - x $

Rewrite: $ y = -x + 8 $

Has a constant term (+8), so not through origin.
➡️ Not direct variation

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5. $ 3x - 6y = 10 $

Solve for $ y $:

$$
-6y = -3x + 10 \\
y = \frac{1}{2}x - \frac{5}{3}
$$

Again, not of the form $ y = kx $ due to constant term.
➡️ Not direct variation

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6. $ -4 + 7x = 8 + 7y $

Solve for $ y $:

$$
7x - 4 = 8 + 7y \\
7x - 12 = 7y \\
y = x - \frac{12}{7}
$$

Has a constant term → not through origin.
➡️ Not direct variation

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## ✔ Problem 2: Identify direct variation from points

Given a set of points:
$ (-3, -1), (-1, -1), (1, 1), (3, 3) $

Let’s check if $ y = kx $ holds.

Check ratios $ \frac{y}{x} $:

- $ \frac{-1}{-3} = \frac{1}{3} $
- $ \frac{-1}{-1} = 1 $
- $ \frac{1}{1} = 1 $
- $ \frac{3}{3} = 1 $

Wait — first point gives $ \frac{1}{3} $, others give 1 → not consistent.

But let’s double-check: Is there a typo?

Wait — point $ (-3, -1) $: $ \frac{y}{x} = \frac{-1}{-3} = \frac{1}{3} $

Point $ (-1, -1) $: $ \frac{-1}{-1} = 1 $

These are different → $ k $ is not constant.

So no, this is not direct variation.

But wait — maybe there’s a mistake in the data? Let’s suppose the first point was meant to be $ (-3, -3) $? Then $ \frac{-3}{-3} = 1 $, and all would have $ k = 1 $. But as written:

→ Not direct variation — inconsistent ratio.

✘ No, these points do not show direct variation.

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## ✔ Problem 3: Tables – Determine if direct variation

We are given tables and asked to see if $ y $ varies directly with $ x $, and if so, find $ k $.

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Table 1:

| x | y |
|---|---|
| 2 | 10 |
| 4 | 20 |

Check $ \frac{y}{x} $:
- $ 10/2 = 5 $
- $ 20/4 = 5 $

Same ratio → $ k = 5 $

✔ Yes, direct variation, $ k = 5 $

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Table 2:

| x | y |
|---|---|
| 3 | 9 |
| 6 | 18 |
| 9 | 27 |

$ y/x $:
- $ 9/3 = 3 $
- $ 18/6 = 3 $
- $ 27/9 = 3 $

✔ Yes, direct variation, $ k = 3 $

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Table 3:

| x | y |
|---|---|
| -2 | 28 |
| 2 | 12 |
| 4 | 18 |

Check $ y/x $:
- $ 28 / (-2) = -14 $
- $ 12 / 2 = 6 $
- $ 18 / 4 = 4.5 $

All different → ✘ Not direct variation

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Table 4:

| x | y |
|---|---|
| 4 | -2.4 |
| 6 | -3.6 |
| 8 | -4.8 |

$ y/x $:
- $ -2.4 / 4 = -0.6 $
- $ -3.6 / 6 = -0.6 $
- $ -4.8 / 8 = -0.6 $

✔ Yes, direct variation, $ k = -0.6 $

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## ✔ Problem 4: Graphs – Is it direct variation?

We are given three graphs. You're supposed to determine if each represents direct variation.

Since I can’t see the image, I'll describe what to look for:

> Direct variation graphs must:
> - Be a straight line
> - Pass through the origin (0, 0)

Let’s analyze each one:

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Graph 1: (Assume linear, passes through origin)

If it's a straight line going through (0,0), then yes.

Example: Line through (0,0), (1,2), (2,4) → $ y = 2x $

✔ Yes, direct variation → Equation: $ y = kx $, where $ k $ is slope.

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Graph 2: (Assume line not through origin)

If it's a straight line but shifted up/down (e.g., intersects y-axis at (0, 2)), then no.

✘ Not direct variation

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Graph 3: (Assume curved or parabolic)

If it's a curve (like a U-shape or V-shape), it’s not linear → not direct variation

✘ No

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Final Answer Format (for graphs):

For each graph:

- Equation? Only if it's direct variation.
- Why? Because it passes through origin and is linear.

Example:
- Graph 1: Equation: $ y = 2x $, Why? It's a straight line through origin.
- Graph 2: No, Why? It doesn’t pass through origin.
- Graph 3: No, Why? It’s not a straight line.

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## ✔ Summary of Answers

Part 1: Equations

| Problem | Is Direct Variation? | Constant $ k $ | Explanation |
|--------|----------------------|------------------|-------------|
| 1 | No | N/A | Has constant term |
| 2 | No | N/A | $ y $ is constant |
| 3 | Yes | $ -2 $ | $ y = -2x $ |
| 4 | No | N/A | Has constant term |
| 5 | No | N/A | Has constant term |
| 6 | No | N/A | Has constant term |

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Part 2: Points

- Points: $ (-3,-1), (-1,-1), (1,1), (3,3) $
- Ratios: $ \frac{y}{x} $ = $ \frac{1}{3}, 1, 1, 1 $ → Not equal
- ✘ Not direct variation

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Part 3: Tables

| Table | Direct Variation? | $ k $ | Reason |
|-------|--------------------|--------|--------|
| 1 | Yes | 5 | $ y/x = 5 $ |
| 2 | Yes | 3 | $ y/x = 3 $ |
| 3 | No | N/A | Ratios differ |
| 4 | Yes | -0.6 | $ y/x = -0.6 $ |

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Part 4: Graphs (Hypothetical)

You need to examine the graphs:

- Graph 1: If straight line through origin → Yes, $ y = kx $
- Graph 2: If line not through origin → No
- Graph 3: If curved → No

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If you can describe the graphs or provide more details, I can give exact answers.

Let me know if you want help filling out the worksheet with boxed answers!