Variation
Math Form 5 · 22 lessons
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## Variation Fundamentals ### Definition Variation refers to the relationship between variables such that changes in one variable correspond to changes in another. There are three main types of variation: - **Direct Variation**: A relationship where one variable is proportional to another. $$y = kx$$ where $k$ is the constant of proportionality. - **Inverse Variation**: A relationship where one variable is inversely proportional to another. $$y = k{x}$$ where $k$ is the constant of proportionality. - **Combined Variation**: A relationship involving both direct and inverse variations. $$y = kx{z}$$ where $k$ is the constant of proportionality. ### Key Concepts - **Direct Variation**: When one variable increases, the other also increases proportionally. - **Inverse Variation**: When one variable increases, the other decreases proportionally. - **Constant of Proportionality**: The fixed number ($k$) that relates two variables in direct or inverse variation. - **Combined Variation**: The simultaneous occurrence of both direct and inverse variation within a relationship. ### Essential Formulas - **Direct Variation**: $$y = kx k = y{x}$$ - **Inverse Variation**: $$y = k{x} k = xy$$ - **Combined Variation**: $$y = kx{z} k = zy{x}$$ ### Basic Algebraic Manipulation (Solving Equations and Inequalities) Algebraic manipulation involves rearranging and simplifying equations or inequalities to isolate variables. - **Solving an equation**: Solve for $x$ in the equation $3x + 7 = 19$. $$3x = 19 - 7 = 12 x = 12{3} = 4$$ - **Solving an inequality**: Solve $2x - 5 < 9$. $$2x < 9 + 5 = 14 x < 14{2} = 7$$ - **Rearranging formulas**: Rearrange $y = mx + c$ to solve for $x$. $$x = y - c{m}, where m 0.$$ ### Function Basics Functions describe the relationship between input ($x$) and output ($y$) such that each input corresponds to exactly one output. - **Direct Function Relationship**: $f(x) = kx$, where $k$ is a constant. - **Inverse Function Relationship**: $f(x) = k{x}$, where $x 0$. - **Composite Functions**: Combining two functions such as $f(x) = 2x$ and $g(x) = x + 3$ to form $f(g(x)) = 2(x + 3)$. ### Examples for Function Basics - **Basic Example**: Given $f(x) = 3x$, find $f(5)$. $$f(5) = 3 5 = 15$$ - **Inverse Function Example**: If $f(x) = 10{x}$, find $f(2)$. $$f(2) = 10{2} = 5$$ ### Core Examples - **Basic example**: If $y$ varies directly as $x$ and $y = 15$ when $x = 3$, find $k$ and the equation. $$k = 15{3} = 5, so y = 5x$$ - **Advanced application**: If $z$ varies directly as $x$ and inversely as $y$, and $z = 10$ when $x = 5$ and $y = 2$, find $k$ and the equation. $$k = 10 2{5} = 4, so z = 4 x{y}$$ ### Related Theorems/Rules - **Proportionality Rule**: In a direct relationship, the ratio of the two variables remains constant. - **Inverse Product Rule**: In an inverse variation, the product of the two variables remains constant. ### Common Pitfalls - Confusing direct and inverse variation when setting up equations. - Forgetting that the constant of proportionality $k$ must be consistent for given conditions. - Misinterpreting the domain of functions in inverse relationships where $x = 0$ is undefined. ### Related Topics - **Linear Equations** - **Graphing Functions** - **Solving Systems of Equations** ### Quick Review Questions - If $y$ varies directly as $x$ and $y = 24$ when $x = 8$, what is $k$? - Given $y$ varies inversely as $x$ and $y = 5$ when $x = 10$, write the equation for this variation. - Solve $3x + 5 = 14$. - Find $f(3)$ if $f(x) = 2x + 1$.
Variations and Functions
If $ z $ varies directly as $ x $ and inversely as $ y $, and $ z = 10 $ when $ x = 5 $ and $ y = 2 $, find $ z $ when $ x = 10 $ and $ y = 4 $.
Why A (10) is correct:
"Direct and inverse" means z = kx/y (where k is a constant). First, find k using the given values: 10 = k(5)/2, so k = 4. Then use z = 4x/y with x = 10 and y = 4: z = 4(10)/4 = 10. ✓
Why the others are wrong:
- B (20): You'd get this if you forgot y is in the denominator and just used z = kx.
- C (8): This comes from incorrectly setting up the relationship or making an arithmetic error.
- D (15): This results from treating the inverse relationship incorrectly in your calculation.
If $ a $ and $ b $ are in the ratio 3:7 and $ b = 21 $, find $ a $.
Why A is correct:
If a:b = 3:7, then a/b = 3/7. Since b = 21, we get a/21 = 3/7, so a = (3/7) × 21 = 9.
Why the others are wrong:
- B (12): This would give a ratio of 12:21 = 4:7, not 3:7.
- C (10): This would give a ratio of 10:21, which doesn't simplify to 3:7.
- D (8): This would give a ratio of 8:21, which also doesn't match 3:7.
If $ y $ varies directly as $ x $, and $ y = 24 $ when $ x = 8 $, find $ y $ when $ x = 12 $.
Why A (36) is correct:
Direct variation means y = kx (where k is constant). First, find k: 24 = k(8), so k = 3. Then use y = 3x: when x = 12, y = 3(12) = 36.
Why the others are wrong:
- B (30): This assumes an incorrect constant of variation (k = 2.5).
- C (32): No valid direct variation formula produces this result.
- D (40): This would come from an incorrect ratio or misapplying the relationship.
If $ y $ varies inversely as $ x $, and $ y = 9 $ when $ x = 3 $, find $ y $ when $ x = 6 $.
Correct answer: A (4.5)
When y varies inversely as x, they follow the rule y = k/x (where k is a constant). First, find k using the given values: 9 = k/3, so k = 27. Now use this to find y when x = 6: y = 27/6 = 4.5.
Why the others are wrong:
- B (6): This would be the answer if y varied *directly* as x (y = kx), not inversely.
- C (3): This matches the original x-value but doesn't follow the inverse relationship.
- D (5): This doesn't come from any correct calculation of the inverse relationship.
Solve for $ x $ in the inequality below: $$ 4x - 7 \leq 13 $$
# Explanation
Why A (5) is correct:
Solve by adding 7 to both sides: 4x ≤ 20, then divide by 4: x ≤ 5. This means x can be 5 or any number smaller than 5, so 5 is the boundary value.
Why the others are wrong:
- B (6): If x = 6, then 4(6) - 7 = 17, which is greater than 13, so it doesn't satisfy the inequality.
- C (4): While 4 works in the inequality, it's not the answer the question is looking for (5 is the boundary/maximum value).
- D (3): This also works in the inequality but isn't the boundary value like 5 is.
If $ y $ varies directly as $ x $, and $ y = 18 $ when $ x = 6 $, find $ y $ when $ x = 15 $.
Why A is correct:
Direct variation means y = kx (where k is a constant). First, find k: 18 = k(6), so k = 3. Then use y = 3x to find y when x = 15: y = 3(15) = 45.
Why the others are wrong:
- B (30): This would be the answer if you incorrectly divided instead of multiplied, or used the wrong ratio.
- C (40): No clear relationship to the direct variation formula.
- D (50): This ignores the constant k and just adds numbers arbitrarily.
If $ y $ varies inversely with $ x $ and $ y = 6 $ when $ x = 3 $, find $ y $ when $ x = 9 $.
# Explanation
Why A (2) is correct:
Inverse variation means y = k/x (where k is a constant). First, find k using y = 6 and x = 3: k = 6 × 3 = 18. Then use y = 18/x to find y when x = 9: y = 18/9 = 2.
Why the others are wrong:
- B (3): This would be correct if the variables had a *direct* relationship, not inverse
- C (4): No clear relationship to the inverse variation formula
- D (6): This assumes y stays the same when x changes, ignoring the inverse relationship
If $ y $ varies directly with $ x^2 $ and $ y = 32 $ when $ x = 4 $, find $ y $ when $ x = 6 $.
Why A (72) is correct:
Direct variation with x² means y = kx². First, find k using y = 32 when x = 4: 32 = k(4²) = 16k, so k = 2. Then when x = 6: y = 2(6²) = 2(36) = 72.
Why the others are wrong:
- B (48): This incorrectly uses y = kx (linear variation) instead of y = kx².
- C (64): This assumes x doubles, so y doubles (32 × 2), ignoring that x² grows faster than x.
- D (96): This results from incorrectly calculating k or the final multiplication.
If $ y $ varies directly with $ x $ and inversely with $ z $, and $ y = 10 $ when $ x = 5 $ and $ z = 2 $, find $ y $ when $ x = 8 $ and $ z = 4 $.
Why A (8) is correct:
When y varies directly with x and inversely with z, the relationship is y = kx/z (where k is a constant). First, find k using the given values: 10 = k(5)/2, so k = 4. Then use this to find y when x = 8 and z = 4: y = 4(8)/4 = 8.
Why the others are wrong:
- B (12): This ignores the inverse relationship with z; if you only account for the direct variation with x, you'd get 12.
- C (10): This assumes y stays the same, but both x and z change, so y must change too.
- D (15): This comes from incorrectly multiplying or miscalculating the constant k.
If $ y $ varies directly with $ x $ and $ y = 15 $ when $ x = 3 $, find $ x $ when $ y = 25 $.
Why A (5) is correct:
Direct variation means y = kx for some constant k. First, find k: 15 = k(3), so k = 5. Then use y = 5x to find x when y = 25: 25 = 5x, so x = 5.
Why the others are wrong:
- B (4): If x = 4, then y = 5(4) = 20, not 25
- C (6): If x = 6, then y = 5(6) = 30, not 25
- D (7): If x = 7, then y = 5(7) = 35, not 25
Solve for $ x $ in the equation $ 3x + 5 = 20 $.
Correct Answer: A (5)
- Subtract 5 from both sides: 3x = 15
- Divide both sides by 3: x = 5
- Check: 3(5) + 5 = 15 + 5 = 20 ✓
Why others are wrong:
- B (15): This is the value of 3x, not x itself
- C (10): This would give 3(10) + 5 = 35, which is too large
- D (4): This would give 3(4) + 5 = 17, which doesn't equal 20
Determine the value of $ k $ if $ 4k - 7 = 13 $.
Correct Answer: A (5)
To solve 4k - 7 = 13, add 7 to both sides to get 4k = 20, then divide by 4 to get k = 5. ✓
Why the others are wrong:
- B (7): If you substitute: 4(7) - 7 = 21, not 13
- C (10): If you substitute: 4(10) - 7 = 33, not 13
- D (4): If you substitute: 4(4) - 7 = 9, not 13
If $ f(x) = 2x - 3 $, find $ f(4) $.
Correct answer: A. 5
To find f(4), substitute x = 4 into the function: f(4) = 2(4) - 3 = 8 - 3 = 5.
Why the others are wrong:
- B (3): This results from 2(4) - 5, an arithmetic error
- C (7): This comes from 2(4) - 1, using the wrong constant
- D (9): This results from 2(4) + 1, adding instead of subtracting
If $ g(x) = \frac{6}{x} $, find $ g(3) $.
Why A is correct:
Substitute x = 3 into the function: g(3) = 6/3 = 2.
Why the others are wrong:
- B (3): This is just the input value; you need to actually divide 6 by 3.
- C (6): This is the numerator, but you still have to divide it by the input.
- D (4): This doesn't result from any correct calculation with these numbers.
If $ x $ varies inversely as $ y $ and $ x = 12 $ when $ y = 2 $, find $ y $ when $ x = 6 $.
Correct answer: A (4)
When x varies inversely as y, they follow the rule xy = k (constant). First, find k using the given values: 12 × 2 = 24. Now use xy = 24 to find y when x = 6: 6y = 24, so y = 4.
Why the others are wrong:
- B (2): This is the original y-value; it doesn't change just because x changed.
- C (6): This would only work if x and y varied *directly* (same ratio), not inversely.
- D (3): This comes from dividing 12 ÷ 4, but doesn't follow the inverse relationship rule.
If $ y $ varies directly as $ x $ and $ y = 18 $ when $ x = 6 $, find $ y $ when $ x = 9 $.
• Direct variation means y = kx (where k is a constant). First, find k: 18 = k(6), so k = 3.
• Now use y = 3x. When x = 9: y = 3(9) = 27 ✓
Why others are wrong:
- B (24): Wrong constant—would need k = 2.67
- C (30): Too high—would need k = 3.33
- D (21): Wrong constant—would need k = 2.33
If $ y $ varies inversely as $ x $ and $ y = 5 $ when $ x = 4 $, find $ x $ when $ y = 10 $.
Why A (2) is correct:
Inverse variation means y = k/x (where k is constant). First, find k: 5 = k/4, so k = 20. Then use y = 20/x to find x when y = 10: 10 = 20/x, so x = 2.
Why the others are wrong:
- B (5): This would give y = 20/5 = 4, not 10
- C (4): This is the original x-value; it doesn't match our new y-value of 10
- D (10): This would give y = 20/10 = 2, not 10
If $ y $ varies directly as $ x $ and inversely as $ z $, and $ y = 12 $ when $ x = 6 $ and $ z = 2 $, find $ y $ when $ x = 9 $ and $ z = 3 $.
Correct Answer: 18
• When y varies directly as x and inversely as z, the formula is: y = kx/z (where k is a constant)
• First, find k using y = 12, x = 6, z = 2: 12 = k(6)/2 → 12 = 3k → k = 4
• Now use y = 4x/z with x = 9 and z = 3: y = 4(9)/3 = 36/3 = 18 ✓
Why others are wrong:
- B (12): This ignores that z changed; if you only consider x changing, you'd incorrectly think y stays the same
- C (15) & D (21): These result from incorrect calculation of k or misapplying the formula
Solve for $ x $ in the inequality $ 5x - 7 > 13 $.
Why A is correct:
Add 7 to both sides: 5x > 20. Then divide by 5: x > 4. The inequality sign stays the same because we're dividing by a positive number.
Why the others are wrong:
- B (x < 4): This has the inequality flipped the wrong way—you only flip it when multiplying/dividing by negatives, which we didn't do.
- C (x > 2): This would be the answer if the original inequality were 5x - 7 > 3 instead.
- D (x < 2): This combines mistakes from both B and C—wrong direction and wrong value.
If $ f(x) = x^2 - 4x + 3 $, find $ f(2) $.
# Explanation
Correct answer: A (-1)
Substitute x = 2 into the function: f(2) = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1 ✓
Why others are wrong:
- B (1): You'd get this if you made an arithmetic error, like forgetting the negative sign on -4(2)
- C (5): This results from incorrectly calculating 4 - 8 + 3 (maybe adding instead of subtracting)
- D (0): You might get 0 if you thought f(2) meant finding where the function equals zero, but we're just evaluating at x = 2
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