Direct and Inverse Variation - Ninth Grade Math
Introduction to Variation
Variation: A relationship between two variables where a change in one variable causes a predictable change in another variable
Constant of Variation (k): A fixed number that relates two variables in a variation equation
Proportionality: When two quantities maintain a constant ratio or product
Constant of Variation (k): A fixed number that relates two variables in a variation equation
Proportionality: When two quantities maintain a constant ratio or product
Two Types of Variation:
1. Direct Variation:
• When one variable increases, the other increases
• When one variable decreases, the other decreases
• Variables move in the SAME direction
• Example: More hours worked = more money earned
2. Inverse Variation:
• When one variable increases, the other decreases
• When one variable decreases, the other increases
• Variables move in OPPOSITE directions
• Example: More workers = less time to complete a job
1. Direct Variation:
• When one variable increases, the other increases
• When one variable decreases, the other decreases
• Variables move in the SAME direction
• Example: More hours worked = more money earned
2. Inverse Variation:
• When one variable increases, the other decreases
• When one variable decreases, the other increases
• Variables move in OPPOSITE directions
• Example: More workers = less time to complete a job
1. Find the Constant of Variation
Constant of Variation (k): The unchanging value that relates two variables in a variation relationship
Also called: Constant of proportionality
Symbol: Usually represented by the letter $k$
Also called: Constant of proportionality
Symbol: Usually represented by the letter $k$
For Direct Variation
Direct Variation Formula:
$$y = kx$$
To find k (constant of variation):
$$k = \frac{y}{x}$$
where $k$ is the constant of variation, and $x \neq 0$
$$y = kx$$
To find k (constant of variation):
$$k = \frac{y}{x}$$
where $k$ is the constant of variation, and $x \neq 0$
Example 1: If $y$ varies directly as $x$, and $y = 12$ when $x = 3$, find the constant of variation.
Solution:
$k = \frac{y}{x} = \frac{12}{3} = 4$
Answer: $k = 4$
The equation is $y = 4x$
Solution:
$k = \frac{y}{x} = \frac{12}{3} = 4$
Answer: $k = 4$
The equation is $y = 4x$
Example 2: Find $k$ if $y = 20$ when $x = 5$ in a direct variation.
Solution:
$k = \frac{20}{5} = 4$
Answer: $k = 4$
Solution:
$k = \frac{20}{5} = 4$
Answer: $k = 4$
For Inverse Variation
Inverse Variation Formula:
$$y = \frac{k}{x} \quad \text{or} \quad xy = k$$
To find k (constant of variation):
$$k = xy$$
where $k$ is the constant of variation
$$y = \frac{k}{x} \quad \text{or} \quad xy = k$$
To find k (constant of variation):
$$k = xy$$
where $k$ is the constant of variation
Example 3: If $y$ varies inversely as $x$, and $y = 8$ when $x = 3$, find the constant of variation.
Solution:
$k = xy = 8 \times 3 = 24$
Answer: $k = 24$
The equation is $y = \frac{24}{x}$ or $xy = 24$
Solution:
$k = xy = 8 \times 3 = 24$
Answer: $k = 24$
The equation is $y = \frac{24}{x}$ or $xy = 24$
Example 4: Find $k$ if $y = 5$ when $x = 12$ in an inverse variation.
Solution:
$k = xy = 5 \times 12 = 60$
Answer: $k = 60$
Solution:
$k = xy = 5 \times 12 = 60$
Answer: $k = 60$
| Variation Type | Formula | Finding k | Example |
|---|---|---|---|
| Direct | $y = kx$ | $k = \frac{y}{x}$ | If $y=12, x=4$, then $k=3$ |
| Inverse | $xy = k$ or $y = \frac{k}{x}$ | $k = xy$ | If $y=12, x=4$, then $k=48$ |
2. Write Direct Variation Equations
Direct Variation: A relationship where $y$ is directly proportional to $x$
Statement: "y varies directly as x" or "y is directly proportional to x"
Notation: $y \propto x$ (y is proportional to x)
Statement: "y varies directly as x" or "y is directly proportional to x"
Notation: $y \propto x$ (y is proportional to x)
General Form of Direct Variation:
$$y = kx$$
where:
• $y$ = dependent variable
• $x$ = independent variable
• $k$ = constant of variation (must be $k \neq 0$)
$$y = kx$$
where:
• $y$ = dependent variable
• $x$ = independent variable
• $k$ = constant of variation (must be $k \neq 0$)
Key Properties of Direct Variation:
• The graph is a straight line through the origin $(0, 0)$
• The ratio $\frac{y}{x}$ is always constant (equals $k$)
• If $x = 0$, then $y = 0$
• The slope of the line is $k$
• All points satisfy: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$
• The graph is a straight line through the origin $(0, 0)$
• The ratio $\frac{y}{x}$ is always constant (equals $k$)
• If $x = 0$, then $y = 0$
• The slope of the line is $k$
• All points satisfy: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$
Steps to Write a Direct Variation Equation:
Step 1: Identify the given point $(x, y)$
Step 2: Find $k$ using $k = \frac{y}{x}$
Step 3: Write the equation as $y = kx$
Step 4: Simplify if needed
Step 1: Identify the given point $(x, y)$
Step 2: Find $k$ using $k = \frac{y}{x}$
Step 3: Write the equation as $y = kx$
Step 4: Simplify if needed
Example 1: Write the direct variation equation if $y = 15$ when $x = 3$.
Step 1: Find $k$
$k = \frac{y}{x} = \frac{15}{3} = 5$
Step 2: Write equation
$y = 5x$
Answer: $y = 5x$
Step 1: Find $k$
$k = \frac{y}{x} = \frac{15}{3} = 5$
Step 2: Write equation
$y = 5x$
Answer: $y = 5x$
Example 2: If $y$ varies directly as $x$, and $y = 24$ when $x = 8$, write the equation.
Solution:
$k = \frac{24}{8} = 3$
Equation: $y = 3x$
Solution:
$k = \frac{24}{8} = 3$
Equation: $y = 3x$
Example 3: The point $(6, 18)$ lies on a direct variation. Write the equation.
Solution:
$k = \frac{18}{6} = 3$
Equation: $y = 3x$
Solution:
$k = \frac{18}{6} = 3$
Equation: $y = 3x$
Example 4: Write a direct variation equation passing through $(-4, -12)$.
Solution:
$k = \frac{-12}{-4} = 3$
Equation: $y = 3x$
Solution:
$k = \frac{-12}{-4} = 3$
Equation: $y = 3x$
Key Points:
• Direct variation equations always pass through the origin
• The constant $k$ can be positive or negative
• If $k > 0$: as $x$ increases, $y$ increases
• If $k < 0$: as $x$ increases, $y$ decreases
• The equation $y = kx$ is in slope-intercept form with $b = 0$
• Direct variation equations always pass through the origin
• The constant $k$ can be positive or negative
• If $k > 0$: as $x$ increases, $y$ increases
• If $k < 0$: as $x$ increases, $y$ decreases
• The equation $y = kx$ is in slope-intercept form with $b = 0$
3. Write and Solve Direct Variation Equations
Solving Direct Variation Problems: Using the direct variation equation to find unknown values
Steps to Write and Solve Direct Variation Equations:
Step 1: Write the direct variation formula: $y = kx$
Step 2: Use given values to find $k$
Step 3: Write the complete equation with the value of $k$
Step 4: Substitute the new value to find the unknown
Step 5: Solve for the variable
Step 1: Write the direct variation formula: $y = kx$
Step 2: Use given values to find $k$
Step 3: Write the complete equation with the value of $k$
Step 4: Substitute the new value to find the unknown
Step 5: Solve for the variable
Alternative Method (Using Proportions):
If $(x_1, y_1)$ and $(x_2, y_2)$ both satisfy a direct variation, then:
$$\frac{y_1}{x_1} = \frac{y_2}{x_2}$$
or cross-multiply:
$$x_1 y_2 = x_2 y_1$$
If $(x_1, y_1)$ and $(x_2, y_2)$ both satisfy a direct variation, then:
$$\frac{y_1}{x_1} = \frac{y_2}{x_2}$$
or cross-multiply:
$$x_1 y_2 = x_2 y_1$$
Example 1: If $y$ varies directly as $x$, and $y = 30$ when $x = 6$, find $y$ when $x = 10$.
Method 1: Using k
Step 1: Find $k = \frac{30}{6} = 5$
Step 2: Equation is $y = 5x$
Step 3: When $x = 10$: $y = 5(10) = 50$
Method 2: Using proportion
$\frac{30}{6} = \frac{y}{10}$
$6y = 300$
$y = 50$
Answer: $y = 50$
Method 1: Using k
Step 1: Find $k = \frac{30}{6} = 5$
Step 2: Equation is $y = 5x$
Step 3: When $x = 10$: $y = 5(10) = 50$
Method 2: Using proportion
$\frac{30}{6} = \frac{y}{10}$
$6y = 300$
$y = 50$
Answer: $y = 50$
Example 2: If $y$ varies directly with $x$, and $y = 12$ when $x = 4$, find $x$ when $y = 30$.
Solution:
Step 1: $k = \frac{12}{4} = 3$
Step 2: Equation: $y = 3x$
Step 3: Substitute $y = 30$: $30 = 3x$
Step 4: Solve: $x = 10$
Answer: $x = 10$
Solution:
Step 1: $k = \frac{12}{4} = 3$
Step 2: Equation: $y = 3x$
Step 3: Substitute $y = 30$: $30 = 3x$
Step 4: Solve: $x = 10$
Answer: $x = 10$
Example 3: The distance a car travels varies directly with time. If the car travels 120 miles in 2 hours, how far will it travel in 5 hours?
Solution:
Let $d$ = distance, $t$ = time
Given: $d = 120$ when $t = 2$
Find $k$: $k = \frac{120}{2} = 60$ mph
Equation: $d = 60t$
When $t = 5$: $d = 60(5) = 300$ miles
Answer: 300 miles
Solution:
Let $d$ = distance, $t$ = time
Given: $d = 120$ when $t = 2$
Find $k$: $k = \frac{120}{2} = 60$ mph
Equation: $d = 60t$
When $t = 5$: $d = 60(5) = 300$ miles
Answer: 300 miles
Example 4: The cost of apples varies directly with the number of pounds. If 3 pounds cost $6, how much will 8 pounds cost?
Solution:
Let $C$ = cost, $p$ = pounds
$k = \frac{6}{3} = 2$ (cost per pound)
Equation: $C = 2p$
When $p = 8$: $C = 2(8) = 16$
Answer: $16
Solution:
Let $C$ = cost, $p$ = pounds
$k = \frac{6}{3} = 2$ (cost per pound)
Equation: $C = 2p$
When $p = 8$: $C = 2(8) = 16$
Answer: $16
4. Identify Direct Variation and Inverse Variation
Identifying Variation Type: Determining whether a relationship is direct, inverse, or neither
Key Concept: Check if the ratio ($\frac{y}{x}$) or product ($xy$) is constant
Key Concept: Check if the ratio ($\frac{y}{x}$) or product ($xy$) is constant
Tests for Variation:
Direct Variation Test:
Calculate $\frac{y}{x}$ for all pairs
• If $\frac{y}{x} = k$ (constant) → Direct Variation
• Equation: $y = kx$
Inverse Variation Test:
Calculate $xy$ for all pairs
• If $xy = k$ (constant) → Inverse Variation
• Equation: $xy = k$ or $y = \frac{k}{x}$
Neither:
If both ratios and products vary → Not a variation
Direct Variation Test:
Calculate $\frac{y}{x}$ for all pairs
• If $\frac{y}{x} = k$ (constant) → Direct Variation
• Equation: $y = kx$
Inverse Variation Test:
Calculate $xy$ for all pairs
• If $xy = k$ (constant) → Inverse Variation
• Equation: $xy = k$ or $y = \frac{k}{x}$
Neither:
If both ratios and products vary → Not a variation
Visual Identification:
Direct Variation:
• Graph: Straight line through origin
• Table: As $x$ increases, $y$ increases (or both decrease)
• $\frac{y}{x}$ is constant
Inverse Variation:
• Graph: Hyperbola (curve)
• Table: As $x$ increases, $y$ decreases (or vice versa)
• $xy$ is constant
Neither:
• Doesn't pass through origin (if linear)
• Neither ratio nor product is constant
Direct Variation:
• Graph: Straight line through origin
• Table: As $x$ increases, $y$ increases (or both decrease)
• $\frac{y}{x}$ is constant
Inverse Variation:
• Graph: Hyperbola (curve)
• Table: As $x$ increases, $y$ decreases (or vice versa)
• $xy$ is constant
Neither:
• Doesn't pass through origin (if linear)
• Neither ratio nor product is constant
Example 1: Identify the type of variation:
Check ratio $\frac{y}{x}$:
$\frac{6}{2} = 3$, $\frac{12}{4} = 3$, $\frac{18}{6} = 3$, $\frac{24}{8} = 3$
All ratios equal 3 (constant)
Answer: Direct Variation with $k = 3$
Equation: $y = 3x$
| x | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| y | 6 | 12 | 18 | 24 |
$\frac{6}{2} = 3$, $\frac{12}{4} = 3$, $\frac{18}{6} = 3$, $\frac{24}{8} = 3$
All ratios equal 3 (constant)
Answer: Direct Variation with $k = 3$
Equation: $y = 3x$
Example 2: Identify the type of variation:
Check product $xy$:
$2 \times 12 = 24$, $3 \times 8 = 24$, $4 \times 6 = 24$, $6 \times 4 = 24$
All products equal 24 (constant)
Answer: Inverse Variation with $k = 24$
Equation: $xy = 24$ or $y = \frac{24}{x}$
| x | 2 | 3 | 4 | 6 |
|---|---|---|---|---|
| y | 12 | 8 | 6 | 4 |
$2 \times 12 = 24$, $3 \times 8 = 24$, $4 \times 6 = 24$, $6 \times 4 = 24$
All products equal 24 (constant)
Answer: Inverse Variation with $k = 24$
Equation: $xy = 24$ or $y = \frac{24}{x}$
Example 3: Determine the variation type:
Check ratio: $\frac{3}{1} = 3$, $\frac{5}{2} = 2.5$, $\frac{7}{3} \approx 2.33$ (not constant)
Check product: $1(3) = 3$, $2(5) = 10$, $3(7) = 21$ (not constant)
Answer: Neither direct nor inverse variation
(This is actually $y = 2x + 1$, a linear equation not passing through origin)
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 3 | 5 | 7 | 9 |
Check product: $1(3) = 3$, $2(5) = 10$, $3(7) = 21$ (not constant)
Answer: Neither direct nor inverse variation
(This is actually $y = 2x + 1$, a linear equation not passing through origin)
Example 4: Identify: $(2, 10)$, $(5, 4)$, $(10, 2)$
Check products:
$2 \times 10 = 20$, $5 \times 4 = 20$, $10 \times 2 = 20$
Answer: Inverse Variation, $k = 20$
Check products:
$2 \times 10 = 20$, $5 \times 4 = 20$, $10 \times 2 = 20$
Answer: Inverse Variation, $k = 20$
| Variation Type | Test | What to Check | Result |
|---|---|---|---|
| Direct | Ratio Test | $\frac{y}{x}$ for all pairs | All ratios are equal |
| Inverse | Product Test | $xy$ for all pairs | All products are equal |
| Neither | Both tests | Check both ratio and product | Neither is constant |
5. Write Inverse Variation Equations
Inverse Variation: A relationship where $y$ is inversely proportional to $x$
Statement: "y varies inversely as x" or "y is inversely proportional to x"
Notation: $y \propto \frac{1}{x}$ (y is inversely proportional to x)
Statement: "y varies inversely as x" or "y is inversely proportional to x"
Notation: $y \propto \frac{1}{x}$ (y is inversely proportional to x)
General Form of Inverse Variation:
$$y = \frac{k}{x} \quad \text{or} \quad xy = k$$
where:
• $y$ = dependent variable
• $x$ = independent variable
• $k$ = constant of variation (must be $k \neq 0$)
• $x \neq 0$ (division by zero is undefined)
$$y = \frac{k}{x} \quad \text{or} \quad xy = k$$
where:
• $y$ = dependent variable
• $x$ = independent variable
• $k$ = constant of variation (must be $k \neq 0$)
• $x \neq 0$ (division by zero is undefined)
Key Properties of Inverse Variation:
• The graph is a hyperbola (curve)
• The product $xy$ is always constant (equals $k$)
• As $x$ increases, $y$ decreases
• As $x$ decreases, $y$ increases
• The curve never touches the axes
• All points satisfy: $x_1 y_1 = x_2 y_2$
• The graph is a hyperbola (curve)
• The product $xy$ is always constant (equals $k$)
• As $x$ increases, $y$ decreases
• As $x$ decreases, $y$ increases
• The curve never touches the axes
• All points satisfy: $x_1 y_1 = x_2 y_2$
Steps to Write an Inverse Variation Equation:
Step 1: Identify the given point $(x, y)$
Step 2: Find $k$ using $k = xy$
Step 3: Write the equation as $y = \frac{k}{x}$ or $xy = k$
Step 4: Simplify if needed
Step 1: Identify the given point $(x, y)$
Step 2: Find $k$ using $k = xy$
Step 3: Write the equation as $y = \frac{k}{x}$ or $xy = k$
Step 4: Simplify if needed
Example 1: Write the inverse variation equation if $y = 8$ when $x = 3$.
Step 1: Find $k$
$k = xy = 3 \times 8 = 24$
Step 2: Write equation
$y = \frac{24}{x}$ or $xy = 24$
Answer: $y = \frac{24}{x}$
Step 1: Find $k$
$k = xy = 3 \times 8 = 24$
Step 2: Write equation
$y = \frac{24}{x}$ or $xy = 24$
Answer: $y = \frac{24}{x}$
Example 2: If $y$ varies inversely as $x$, and $y = 15$ when $x = 4$, write the equation.
Solution:
$k = xy = 4 \times 15 = 60$
Equation: $y = \frac{60}{x}$ or $xy = 60$
Solution:
$k = xy = 4 \times 15 = 60$
Equation: $y = \frac{60}{x}$ or $xy = 60$
Example 3: The point $(5, 12)$ lies on an inverse variation. Write the equation.
Solution:
$k = 5 \times 12 = 60$
Equation: $y = \frac{60}{x}$
Solution:
$k = 5 \times 12 = 60$
Equation: $y = \frac{60}{x}$
Example 4: Write an inverse variation equation passing through $(10, 2)$.
Solution:
$k = 10 \times 2 = 20$
Equation: $y = \frac{20}{x}$
Solution:
$k = 10 \times 2 = 20$
Equation: $y = \frac{20}{x}$
Key Points:
• Inverse variation equations never pass through the origin
• The constant $k$ can be positive or negative
• If $k > 0$: curve is in Quadrants I and III
• If $k < 0$: curve is in Quadrants II and IV
• The equation can be written as $y = \frac{k}{x}$ or $xy = k$
• Inverse variation equations never pass through the origin
• The constant $k$ can be positive or negative
• If $k > 0$: curve is in Quadrants I and III
• If $k < 0$: curve is in Quadrants II and IV
• The equation can be written as $y = \frac{k}{x}$ or $xy = k$
6. Write and Solve Inverse Variation Equations
Solving Inverse Variation Problems: Using the inverse variation equation to find unknown values
Steps to Write and Solve Inverse Variation Equations:
Step 1: Write the inverse variation formula: $y = \frac{k}{x}$ or $xy = k$
Step 2: Use given values to find $k$
Step 3: Write the complete equation with the value of $k$
Step 4: Substitute the new value to find the unknown
Step 5: Solve for the variable
Step 1: Write the inverse variation formula: $y = \frac{k}{x}$ or $xy = k$
Step 2: Use given values to find $k$
Step 3: Write the complete equation with the value of $k$
Step 4: Substitute the new value to find the unknown
Step 5: Solve for the variable
Alternative Method (Using Products):
If $(x_1, y_1)$ and $(x_2, y_2)$ both satisfy an inverse variation, then:
$$x_1 y_1 = x_2 y_2$$
or rearrange:
$$\frac{x_1}{x_2} = \frac{y_2}{y_1}$$
If $(x_1, y_1)$ and $(x_2, y_2)$ both satisfy an inverse variation, then:
$$x_1 y_1 = x_2 y_2$$
or rearrange:
$$\frac{x_1}{x_2} = \frac{y_2}{y_1}$$
Example 1: If $y$ varies inversely as $x$, and $y = 10$ when $x = 6$, find $y$ when $x = 15$.
Method 1: Using k
Step 1: Find $k = xy = 6 \times 10 = 60$
Step 2: Equation is $y = \frac{60}{x}$
Step 3: When $x = 15$: $y = \frac{60}{15} = 4$
Method 2: Using product rule
$x_1 y_1 = x_2 y_2$
$6(10) = 15(y)$
$60 = 15y$
$y = 4$
Answer: $y = 4$
Method 1: Using k
Step 1: Find $k = xy = 6 \times 10 = 60$
Step 2: Equation is $y = \frac{60}{x}$
Step 3: When $x = 15$: $y = \frac{60}{15} = 4$
Method 2: Using product rule
$x_1 y_1 = x_2 y_2$
$6(10) = 15(y)$
$60 = 15y$
$y = 4$
Answer: $y = 4$
Example 2: If $y$ varies inversely with $x$, and $y = 8$ when $x = 5$, find $x$ when $y = 10$.
Solution:
Step 1: $k = xy = 5 \times 8 = 40$
Step 2: Equation: $xy = 40$
Step 3: Substitute $y = 10$: $x(10) = 40$
Step 4: Solve: $x = 4$
Answer: $x = 4$
Solution:
Step 1: $k = xy = 5 \times 8 = 40$
Step 2: Equation: $xy = 40$
Step 3: Substitute $y = 10$: $x(10) = 40$
Step 4: Solve: $x = 4$
Answer: $x = 4$
Example 3: The time to complete a job varies inversely with the number of workers. If 4 workers can complete a job in 12 hours, how long will it take 6 workers?
Solution:
Let $t$ = time, $w$ = number of workers
Given: $t = 12$ when $w = 4$
Find $k$: $k = wt = 4 \times 12 = 48$ worker-hours
Equation: $wt = 48$
When $w = 6$: $6t = 48$, so $t = 8$ hours
Answer: 8 hours
Solution:
Let $t$ = time, $w$ = number of workers
Given: $t = 12$ when $w = 4$
Find $k$: $k = wt = 4 \times 12 = 48$ worker-hours
Equation: $wt = 48$
When $w = 6$: $6t = 48$, so $t = 8$ hours
Answer: 8 hours
Example 4: The speed of a car is inversely proportional to the time it takes to travel a fixed distance. If the car travels at 60 mph and takes 3 hours, how fast must it go to make the trip in 2 hours?
Solution:
Let $s$ = speed, $t$ = time
$k = st = 60 \times 3 = 180$ miles
Equation: $st = 180$
When $t = 2$: $s(2) = 180$, so $s = 90$ mph
Answer: 90 mph
Solution:
Let $s$ = speed, $t$ = time
$k = st = 60 \times 3 = 180$ miles
Equation: $st = 180$
When $t = 2$: $s(2) = 180$, so $s = 90$ mph
Answer: 90 mph
Example 5: The pressure of a gas varies inversely with its volume. If the pressure is 20 psi when the volume is 15 cubic feet, what is the pressure when the volume is 10 cubic feet?
Solution:
$k = pv = 20 \times 15 = 300$
When $v = 10$: $p(10) = 300$
$p = 30$ psi
Answer: 30 psi
Solution:
$k = pv = 20 \times 15 = 300$
When $v = 10$: $p(10) = 300$
$p = 30$ psi
Answer: 30 psi
Direct vs. Inverse Variation Comparison
| Feature | Direct Variation | Inverse Variation |
|---|---|---|
| Equation | $y = kx$ | $y = \frac{k}{x}$ or $xy = k$ |
| Finding k | $k = \frac{y}{x}$ | $k = xy$ |
| Relationship | As x increases, y increases | As x increases, y decreases |
| Graph | Straight line through origin | Hyperbola (curve) |
| Constant | $\frac{y}{x}$ = constant | $xy$ = constant |
| Proportion | $\frac{y_1}{x_1} = \frac{y_2}{x_2}$ | $x_1 y_1 = x_2 y_2$ |
| Example | Distance = speed × time (at constant speed) | Time = distance ÷ speed (at constant distance) |
Quick Reference Guide
Direct Variation Formulas:
• Equation: $y = kx$
• Find k: $k = \frac{y}{x}$
• Check: $\frac{y}{x}$ = constant
• Proportion: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$
• Equation: $y = kx$
• Find k: $k = \frac{y}{x}$
• Check: $\frac{y}{x}$ = constant
• Proportion: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$
Inverse Variation Formulas:
• Equation: $y = \frac{k}{x}$ or $xy = k$
• Find k: $k = xy$
• Check: $xy$ = constant
• Proportion: $x_1 y_1 = x_2 y_2$
• Equation: $y = \frac{k}{x}$ or $xy = k$
• Find k: $k = xy$
• Check: $xy$ = constant
• Proportion: $x_1 y_1 = x_2 y_2$
Identifying Variation Type:
For Direct Variation:
1. Check if $\frac{y}{x}$ is constant
2. Graph passes through origin
3. Both variables move in same direction
For Inverse Variation:
1. Check if $xy$ is constant
2. Graph is a hyperbola
3. Variables move in opposite directions
For Direct Variation:
1. Check if $\frac{y}{x}$ is constant
2. Graph passes through origin
3. Both variables move in same direction
For Inverse Variation:
1. Check if $xy$ is constant
2. Graph is a hyperbola
3. Variables move in opposite directions
Problem-Solving Strategy:
1. Identify the type of variation
2. Write the appropriate formula
3. Use given values to find $k$
4. Write the complete equation
5. Substitute new values to solve
6. Check your answer makes sense
1. Identify the type of variation
2. Write the appropriate formula
3. Use given values to find $k$
4. Write the complete equation
5. Substitute new values to solve
6. Check your answer makes sense
Real-World Applications:
Direct Variation Examples:
• Cost vs. quantity purchased
• Distance vs. time (at constant speed)
• Earnings vs. hours worked
• Perimeter of square vs. side length
Inverse Variation Examples:
• Speed vs. time (for fixed distance)
• Number of workers vs. time to complete job
• Pressure vs. volume (Boyle's Law)
• Length vs. width (for fixed area)
Direct Variation Examples:
• Cost vs. quantity purchased
• Distance vs. time (at constant speed)
• Earnings vs. hours worked
• Perimeter of square vs. side length
Inverse Variation Examples:
• Speed vs. time (for fixed distance)
• Number of workers vs. time to complete job
• Pressure vs. volume (Boyle's Law)
• Length vs. width (for fixed area)
Success Tips for Variation Problems:
✓ Identify whether it's direct or inverse variation first
✓ For direct: check if ratio $\frac{y}{x}$ is constant
✓ For inverse: check if product $xy$ is constant
✓ Always find $k$ before writing the equation
✓ Remember: direct variation passes through origin, inverse does not
✓ Use units to verify your answer makes sense
✓ Practice recognizing real-world variation relationships
✓ Double-check by substituting back into the equation
✓ Identify whether it's direct or inverse variation first
✓ For direct: check if ratio $\frac{y}{x}$ is constant
✓ For inverse: check if product $xy$ is constant
✓ Always find $k$ before writing the equation
✓ Remember: direct variation passes through origin, inverse does not
✓ Use units to verify your answer makes sense
✓ Practice recognizing real-world variation relationships
✓ Double-check by substituting back into the equation