Clear educational guide on direct and inverse variation, including formulas, examples, and methods to find the constant of variation (k).

Let's solve each example step by step and explain the solutions clearly.

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1. Direct Variation

Definition:
- The equation is in the form:
$$
y = kx
$$
- We use:
$$
\frac{y_1}{x_1} = \frac{y_2}{x_2}
$$
to find a missing $ x $ or $ y $.
- To find the constant of variation $ k $, we use:
$$
k = \frac{y}{x}
$$

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#### Example 1:
> $ y $ varies directly with $ x $. If $ x $ is 5 when $ y $ is 7, what is $ x $ when $ y $ is 14?

Step 1: Use the proportion:
$$
\frac{y_1}{x_1} = \frac{y_2}{x_2}
$$
Plug in known values:
$$
\frac{7}{5} = \frac{14}{x_2}
$$

Step 2: Solve for $ x_2 $:
Cross-multiply:
$$
7x_2 = 5 \times 14 = 70
\Rightarrow x_2 = \frac{70}{7} = 10
$$

✔ Answer: $ x = 10 $

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#### Example 2:
> $ y $ varies directly with $ x $. If $ x $ is 10 when $ y $ is 5, what is the constant of variation $ k $?

Use:
$$
k = \frac{y}{x} = \frac{5}{10} = 0.5
$$

✔ Answer: $ k = 0.5 $

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2. Inverse Variation

Definition:
- The equation is in the form:
$$
y = \frac{k}{x}
$$
- We use:
$$
x_1 y_1 = x_2 y_2
$$
to find a missing $ x $ or $ y $.
- To find the constant of variation $ k $, use:
$$
k = x \cdot y
$$

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#### Example 3:
> $ y $ varies inversely with $ x $. If $ x $ is 5 when $ y $ is 7, what is $ x $ when $ y $ is 14?

Use:
$$
x_1 y_1 = x_2 y_2
$$
$$
5 \times 7 = x_2 \times 14
\Rightarrow 35 = 14x_2
\Rightarrow x_2 = \frac{35}{14} = 2.5
$$

✔ Answer: $ x = 2.5 $

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#### Example 4:
> $ y $ varies inversely with $ x $. If $ x $ is 2 when $ y $ is 7, what is the constant of variation $ k $?

Use:
$$
k = x \cdot y = 2 \times 7 = 14
$$

✔ Answer: $ k = 14 $

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✔ Final Answers Summary:

| Example | Question | Answer |
|--------|---------|--------|
| 1 | If $ x=5 $ when $ y=7 $, find $ x $ when $ y=14 $ (direct) | $ x = 10 $ |
| 2 | If $ x=10 $, $ y=5 $, find $ k $ (direct) | $ k = 0.5 $ |
| 3 | If $ x=5 $, $ y=7 $, find $ x $ when $ y=14 $ (inverse) | $ x = 2.5 $ |
| 4 | If $ x=2 $, $ y=7 $, find $ k $ (inverse) | $ k = 14 $ |

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🔍 Key Takeaways:

- Direct Variation: As one variable increases, the other increases proportionally. $ y = kx $
- Inverse Variation: As one increases, the other decreases. $ y = \frac{k}{x} $
- Always use $ k = \frac{y}{x} $ (direct) or $ k = xy $ (inverse) to find the constant first if needed.
- Use proportions or product equality to find missing values.

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