Direct, Inverse, and Joint Variation Worksheet with problems and equations.

Let’s solve each problem step by step, following the worksheet’s instructions.

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## Part 1: State whether each equation represents a direct, inverse, or joint variation. Name the constant of variation.

1) \( y = 2x \)

- This is direct variation because \( y \) is proportional to \( x \).
- General form: \( y = kx \), so \( k = 2 \).
✔ Answer: Direct variation; constant of variation = 2

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2) \( \frac{x}{5} = y \) → rewritten as \( y = \frac{1}{5}x \)

- Again, this is direct variation.
- Constant of variation \( k = \frac{1}{5} \).
✔ Answer: Direct variation; constant of variation = \( \frac{1}{5} \)

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3) \( xy = 12 \) → rewritten as \( y = \frac{12}{x} \)

- This is inverse variation because \( y \) is inversely proportional to \( x \).
- General form: \( y = \frac{k}{x} \), so \( k = 12 \).
✔ Answer: Inverse variation; constant of variation = 12

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4) \( D = \frac{3}{4}gh \)

- Here, \( D \) varies with two variables: \( g \) and \( h \). Since it’s proportional to the product of two variables, this is joint variation.
- General form: \( D = kgh \), so \( k = \frac{3}{4} \).
✔ Answer: Joint variation; constant of variation = \( \frac{3}{4} \)

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## Part 2: Translate each statement into a formula. Use \( k \) as the constant of variation.

5) E varies jointly as M and the square of V.

- “Varies jointly” means proportional to the product.
- Square of V = \( V^2 \)
✔ Formula: \( E = k \cdot M \cdot V^2 \)

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6) The volume, V, of a gas varies directly as the temperature, T, and inversely as the pressure P.

- Directly as T → multiply by T
- Inversely as P → divide by P
✔ Formula: \( V = k \cdot \frac{T}{P} \)

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7) The mass, M, of a cement block varies jointly as the length, L, width, W, and thickness, T, of the block.

- Joint variation with three variables → multiply all together
✔ Formula: \( M = k \cdot L \cdot W \cdot T \)

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8) P varies directly as the square of V and inversely as R.

- Directly as square of V → multiply by \( V^2 \)
- Inversely as R → divide by R
✔ Formula: \( P = k \cdot \frac{V^2}{R} \)

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## Part 3: Write an equation for each statement. Then, solve the equation.

9) If y varies inversely as x and y = 2 when x = 8, find x when y = 14.

Step 1: General formula
Inverse variation → \( y = \frac{k}{x} \)

Step 2: Find k using given values
When \( x = 8 \), \( y = 2 \):
\[
2 = \frac{k}{8} \Rightarrow k = 2 \cdot 8 = 16
\]

Step 3: Rewrite formula with k
\[
y = \frac{16}{x}
\]

Step 4: Solve for x when y = 14
\[
14 = \frac{16}{x} \Rightarrow x = \frac{16}{14} = \frac{8}{7}
\]

✔ Answer: \( x = \frac{8}{7} \)

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10) Suppose y varies jointly with x and z. If y = 20 when x = 2 and z = 5, find y when x = 14 and z = 8.

Step 1: General formula
Joint variation → \( y = kxz \)

Step 2: Find k
Given: \( y = 20 \), \( x = 2 \), \( z = 5 \)
\[
20 = k \cdot 2 \cdot 5 = 10k \Rightarrow k = 2
\]

Step 3: Rewrite formula
\[
y = 2xz
\]

Step 4: Plug in x = 14, z = 8
\[
y = 2 \cdot 14 \cdot 8 = 224
\]

✔ Answer: \( y = 224 \)

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11) If y varies inversely as x and x = 7 when y = 21, find y when x = 42.

Step 1: Formula
\( y = \frac{k}{x} \)

Step 2: Find k
When \( x = 7 \), \( y = 21 \):
\[
21 = \frac{k}{7} \Rightarrow k = 21 \cdot 7 = 147
\]

Step 3: Formula becomes
\[
y = \frac{147}{x}
\]

Step 4: Find y when x = 42
\[
y = \frac{147}{42} = \frac{49}{14} = \frac{7}{2} = 3.5
\]

✔ Answer: \( y = \frac{7}{2} \) or 3.5

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12) Find y when x = 1.5, if y varies directly as x and y = -16 when x = 6.

Step 1: Formula
Direct variation → \( y = kx \)

Step 2: Find k
When \( x = 6 \), \( y = -16 \):
\[
-16 = k \cdot 6 \Rightarrow k = \frac{-16}{6} = -\frac{8}{3}
\]

Step 3: Formula becomes
\[
y = -\frac{8}{3}x
\]

Step 4: Plug in x = 1.5 (which is \( \frac{3}{2} \))
\[
y = -\frac{8}{3} \cdot \frac{3}{2} = -\frac{24}{6} = -4
\]

✔ Answer: \( y = -4 \)

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## ✔ Final Answers Summary:

Part 1:

1) Direct; k=2
2) Direct; k=1/5
3) Inverse; k=12
4) Joint; k=3/4

Part 2:

5) \( E = k M V^2 \)
6) \( V = k \frac{T}{P} \)
7) \( M = k L W T \)
8) \( P = k \frac{V^2}{R} \)

Part 3:

9) \( x = \frac{8}{7} \)
10) \( y = 224 \)
11) \( y = \frac{7}{2} \)
12) \( y = -4 \)

Let me know if you’d like these explained visually or with graphs!