Here is the complete solution to the
Direct Variation Worksheet, with clear explanations for each problem.
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## 🔹 Key Concept: Direct Variation
A direct variation is a relationship between two variables where one is a constant multiple of the other. It is written as:
>
y = kx
Where:
- `y` and `x` are variables
- `k` is the
constant of variation (a non-zero constant)
- The graph is a straight line passing through the origin (0,0)
If an equation has a
+ or – constant term (like y = mx + b where b ≠ 0), it is
NOT direct variation.
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##
✔ Problem 1: Which equation is *not* an example of direct variation?
A. y = -7/3 x + 1
B. y = 5/16 x
C. y = 4x
D. y = -9x
✔ Answer: A
Explanation:
Only
A has a "+1" — this means the line does not pass through the origin. All others are in the form y = kx.
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##
✔ Problem 2: Which equation is *not* an example of direct variation?
A. y = x → y = 1·x → ✔️ direct variation
B. 2x + 3y = 0 → Solve for y: 3y = -2x → y = -2/3 x → ✔️ direct variation
C. y = 1/2 x → ✔️ direct variation
D. 5x + 6y = 30 → Solve for y: 6y = -5x + 30 → y = (-5/6)x + 5 →
✘ NOT direct variation (has +5)
✔ Answer: D
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##
✔ Problems 3–5: Name the constant of variation (k) for each equation.
3. y = 5x
→ k =
5
4. y = (1/2)x
→ k =
1/2
5. y = (-2/3)x
→ k =
-2/3
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##
✔ Problems 6–10: Write a direct variation equation and solve.
6. Suppose y varies directly as x, and y = 16 when x = 8. Find y when x = 16.
Step 1: Write general form:
y = kx
Step 2: Plug in given values to find k:
16 = k · 8 → k = 16 / 8 =
2
Step 3: Equation:
y = 2x
Step 4: Find y when x = 16:
y = 2 · 16 =
32
✔ Answer: y = 32
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7. Suppose y varies directly as x, and y = 21 when x = 3. Find x when y = 42.
Step 1: y = kx
21 = k · 3 → k = 21 / 3 =
7
Step 2: Equation:
y = 7x
Step 3: Find x when y = 42:
42 = 7x → x = 42 / 7 =
6
✔ Answer: x = 6
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8. Suppose v varies directly as g, and v = 36 when g = 4. Find v when g = 11.
Step 1: v = kg
36 = k · 4 → k = 36 / 4 =
9
Step 2: Equation:
v = 9g
Step 3: Find v when g = 11:
v = 9 · 11 =
99
✔ Answer: v = 99
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9. Suppose a varies directly as b, and a = 7 when b = 2. Find b when a = 21.
Step 1: a = kb
7 = k · 2 → k = 7 / 2 =
3.5 or
7/2
Step 2: Equation:
a = (7/2)b
Step 3: Find b when a = 21:
21 = (7/2)b → Multiply both sides by 2: 42 = 7b → b = 42 / 7 =
6
✔ Answer: b = 6
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10. Suppose y varies directly as x, and y = 9 when x = 3/2. Find y when x = 1.
Step 1: y = kx
9 = k · (3/2) → k = 9 ÷ (3/2) = 9 × (2/3) =
6
Step 2: Equation:
y = 6x
Step 3: Find y when x = 1:
y = 6 · 1 =
6
✔ Answer: y = 6
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## 📝 Final Answers Summary:
1.
A
2.
D
3.
k = 5
4.
k = 1/2
5.
k = -2/3
6.
y = 32
7.
x = 6
8.
v = 99
9.
b = 6
10.
y = 6
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Parent Tip: Review the logic above to help your child master the concept of direct variation practice worksheet.
