Direct Variation

A direct variation is a relationship between two variables in which one variable changes in direct proportion to the other.

For example, suppose apples cost $3 per kilogram.

If x represents the mass of apples in kilograms and y represents the total cost:

y = 3x

If you buy twice as many kilograms, the cost doubles. If you buy three times as many, the cost triples.

This is a direct variation because the ratio between y and x remains constant.

---

The Direct Variation Equation

A direct variation can be written in the form:

y = kx

where:

• x = independent variable
• y = dependent variable
• k = constant of variation

The value k tells us how much y changes for each unit of x.

For example:

y = 5x

has:

k = 5

This means y is always 5 times x.

---

The Constant of Variation

The constant of variation is the constant ratio between the two variables.

From:

y = kx

divide both sides by x:

k = y/x

Therefore, for a direct variation:

y/x = constant

For example, suppose:

x = 4

and:

y = 20

Then:

k = 20/4

k = 5

The direct variation equation is:

y = 5x

---

Understanding the Constant

Suppose:

d = 60t

where:

d = distance in kilometres

t = time in hours

The constant of variation is:

60

In context, this means:

60 km/h

The constant of variation often represents a real-world rate such as:

• price per item
• speed
• pay per hour
• mass per unit
• distance per unit time
• amount used per person

---

Direct Variation and Proportional Relationships

A direct variation is also called a proportional relationship.

If:

y = kx

then:

y/x = k

The ratio between the two quantities remains constant.

For example:

Calculate y/x:

4/1 = 4

8/2 = 4

12/3 = 4

16/4 = 4

20/5 = 4

The ratio is always:

4

Therefore:

k = 4

and:

y = 4x

---

Recognizing Direct Variation from a Table

Consider:

Calculate:

y/x

6/2 = 3

12/4 = 3

18/6 = 3

24/8 = 3

Because the ratio is constant:

k = 3

Therefore:

y = 3x

This table represents direct variation.

---

A Table That Is Not Direct Variation

Consider:

The y-values increase by 3 each time, so the relationship is linear.

However:

5/1 = 5

8/2 = 4

11/3 ≠ 4

14/4 = 3.5

The ratio y/x is not constant.

Therefore:

This is linear, but it is not a direct variation.

Its equation is:

y = 3x + 2

The +2 prevents it from being a direct variation.

---

Direct Variation Is a Special Linear Relationship

All direct variations are linear relationships.

However:

not all linear relationships are direct variations.

A general linear equation is:

y = mx + b

A direct variation has:

b = 0

Therefore:

y = kx

A direct variation has no non-zero starting value.

---

The Origin

When x = 0 in a direct variation:

y = k(0)

y = 0

Therefore, every direct variation contains the point:

(0, 0)

This point is called the origin.

A graph representing direct variation must pass through the origin.

---

Direct Variation on a Graph

Consider:

y = 2x

Some points are:

The graph:

• is a straight line
• passes through (0, 0)
• has a constant slope
• represents a proportional relationship

The slope is the constant of variation:

k = 2

---

Slope and the Constant of Variation

For a general linear equation:

y = mx + b

m represents slope.

For direct variation:

y = kx

therefore:

m = k

The slope and the constant of variation are the same value.

For example:

y = −4x

has:

slope = −4

constant of variation = −4

---

Positive Direct Variation

Consider:

y = 3x

As x increases, y increases.

If x doubles:

y doubles.

If x triples:

y triples.

For example:

x = 2 → y = 6

x = 4 → y = 12

x = 6 → y = 18

This is a positive direct variation.

---

Negative Direct Variation

Direct variation can also have a negative constant.

Consider:

y = −2x

If x increases:

y decreases.

For example:

x = 1 → y = −2

x = 2 → y = −4

x = 3 → y = −6

The ratio remains:

y/x = −2

Therefore, it is still a direct variation.

---

Finding k from One Pair of Values

Suppose y varies directly with x.

When:

x = 6

y = 24

Use:

k = y/x

k = 24/6

k = 4

Therefore:

y = 4x

Once the equation is known, we can determine any corresponding value.

---

Finding an Unknown Output

Suppose:

y varies directly with x

and:

y = 18 when x = 3.

First find k:

k = 18/3

k = 6

Therefore:

y = 6x

Now find y when x = 8:

y = 6(8)

y = 48

---

Finding an Unknown Input

Suppose:

y = 7x

and:

y = 63

Substitute:

63 = 7x

Divide by 7:

x = 9

Therefore, an output of 63 corresponds to an input of 9.

---

Solving with Proportions

Because direct variation has a constant ratio:

y/x = k

two corresponding pairs satisfy:

y₁/x₁ = y₂/x₂

For example, if:

x = 3 when y = 12

find y when x = 8.

Write:

12/3 = y/8

4 = y/8

Multiply by 8:

y = 32

This gives the same result as finding k first.

---

Two Methods for Direct Variation Problems

Suppose 5 notebooks cost $15. Find the cost of 8 notebooks.

Method 1: Find k

k = 15/5

k = 3

Equation:

C = 3n

For 8 notebooks:

C = 3(8)

C = $24

Method 2: Use a proportion

15/5 = C/8

3 = C/8

C = $24

Both methods use the same proportional relationship.

---

Scaling Up

If y varies directly with x, multiplying x by a factor multiplies y by the same factor.

Suppose:

y = 4x

If x changes from 3 to 12:

12 is:

4 times 3

Therefore, y also becomes four times larger.

At x = 3:

y = 12

At x = 12:

y = 48

---

Scaling Down

Suppose:

y = 10x

If x is divided by 2, y is also divided by 2.

For example:

x = 8 → y = 80

x = 4 → y = 40

This proportional scaling is a defining feature of direct variation.

---

Real-World Example: Hourly Pay

Suppose a worker earns:

$18 per hour

with no fixed bonus.

Let:

h = hours worked

P = total pay

Then:

P = 18h

The constant of variation is:

18 dollars/hour

If the worker works 6 hours:

P = 18(6)

P = $108

---

Why a Fixed Bonus Changes the Relationship

Suppose the worker instead receives:

$18 per hour plus a $25 bonus

Then:

P = 18h + 25

This is linear.

However, it is not direct variation because:

P ≠ 0 when h = 0.

The graph does not pass through the origin.

This distinction is important.

---

Real-World Example: Constant Speed

Suppose a cyclist travels at a constant speed of:

15 km/h

Distance is:

d = 15t

where:

d = distance in kilometres

t = time in hours

The constant of variation is:

15 km/h

After 4 hours:

d = 15(4)

d = 60 km

Distance varies directly with time when speed remains constant.

---

Real-World Example: Cost per Item

Suppose pens cost:

$2.50 each

Let:

n = number of pens

C = total cost

Then:

C = 2.5n

The constant of variation is:

$2.50 per pen

For 12 pens:

C = 2.5(12)

C = $30

---

Real-World Example: Recipe Quantities

A recipe requires:

250 g of flour for each batch

Let:

b = number of batches

F = flour required in grams

Then:

F = 250b

For 3 batches:

F = 250(3)

F = 750 g

For half a batch:

F = 250(0.5)

F = 125 g

The amount of flour varies directly with the number of batches.

---

Real-World Example: Mass and Number of Objects

Suppose identical metal blocks each have a mass of:

2.4 kg

Let:

n = number of blocks

M = total mass

Then:

M = 2.4n

The constant of variation is:

2.4 kg/block

If there are 15 blocks:

M = 2.4(15)

M = 36 kg

---

Real-World Example: Distance on a Map

Suppose a map uses the scale:

1 cm = 5 km

Let:

m = map distance in centimetres

D = actual distance in kilometres

Then:

D = 5m

The constant of variation is:

5 km/cm

If two locations are 7.5 cm apart on the map:

D = 5(7.5)

D = 37.5 km

---

Real-World Example: Currency Conversion

Suppose, for a simplified example, one unit of currency A converts to 1.4 units of currency B at a fixed rate.

Let:

A = amount in currency A

B = amount in currency B

Then:

B = 1.4A

The constant of variation is:

1.4

If A = 50:

B = 1.4(50)

B = 70

Actual exchange rates can change, but at a fixed conversion rate the relationship is proportional.

---

Real-World Example: Density

For a material with constant density:

mass = density × volume

Suppose a material has density:

2.7 g/cm³

Then:

m = 2.7V

where:

m = mass in grams

V = volume in cm³

Mass varies directly with volume.

The constant of variation is the density:

k = 2.7 g/cm³

For V = 20 cm³:

m = 2.7(20)

m = 54 g

---

Real-World Example: Distance and Time

At constant speed:

d = vt

If the speed is fixed at:

v = 12 m/s

then:

d = 12t

This is direct variation between distance and time.

The constant of variation is:

12 m/s

Notice that the familiar science equation becomes a direct variation when one quantity, in this case speed, remains constant.

---

Recognizing Direct Variation from Words

Look for phrases such as:

• "per"
• "for each"
• "at a constant rate"
• "proportional to"
• "varies directly with"
• "for every"

For example:

"A printer produces 25 pages per minute."

If p is pages and t is time:

p = 25t

This is direct variation.

---

But "Per" Does Not Always Guarantee Direct Variation

Suppose a taxi charges:

$4 per kilometre plus a $6 starting fee

The cost equation is:

C = 4d + 6

Although the situation contains "$4 per kilometre," there is also a non-zero starting value.

Therefore:

This is not direct variation.

Always check whether the relationship has the form:

y = kx

---

Identifying Direct Variation from an Equation

Which equations represent direct variation?

y = 5x

Yes.

y = −3x

Yes.

y = 0.4x

Yes.

y = 5x + 2

No.

y = 7 − 3x

No.

y = x²

No.

A direct variation must be expressible as:

y = kx

---

Identifying Direct Variation from a Graph

A graph represents direct variation when it:

• is a straight line
• passes through the origin (0, 0)

Both conditions are necessary.

A straight line that crosses the y-axis somewhere other than zero is linear but not proportional.

---

Direct Variation vs Other Linear Relationships

Compare:

A: y = 4x

and:

B: y = 4x + 6

Both have slope:

4

Both are linear.

But only A is direct variation.

For A:

when x = 0, y = 0.

For B:

when x = 0, y = 6.

Therefore, B has a non-zero intercept and is not directly proportional.

---

Constant Difference vs Constant Ratio

This distinction is important.

A linear relationship has a constant rate of change.

A direct variation also has a constant ratio y/x.

Consider:

y = 3x + 5

Its slope is constantly 3.

But:

y/x

does not remain constant.

Therefore, the relationship is linear but not proportional.

For:

y = 3x

both the rate of change and y/x remain 3.

Therefore, it is direct variation.

---

Building an Equation from a Table

Consider:

Calculate:

21/3 = 7

35/5 = 7

56/8 = 7

70/10 = 7

Therefore:

k = 7

Equation:

y = 7x

---

Checking the Equation

For:

y = 7x

use x = 8:

y = 7(8)

y = 56

The table contains:

(8, 56)

The equation correctly represents the relationship.

Checking several pairs provides stronger verification.

---

Finding a Missing Table Value

Suppose:

From:

10/2 = 5

the constant is:

k = 5

Therefore:

y = 5x

For x = 7:

y = 5(7)

y = 35

---

Finding a Missing Input

Suppose:

y = 8x

and the output is:

y = 72

Then:

72 = 8x

x = 9

Therefore:

x = 9

---

Comparing Two Direct Variations

Consider:

A: y = 3x

B: y = 5x

Both are direct variations.

For A:

k = 3

For B:

k = 5

For the same positive input, B produces a larger output because its constant of variation is larger.

---

Direct Variation and Unit Rates

The constant of variation often represents a unit rate.

Suppose:

4 kg of fruit costs $14.

Find the cost per kilogram:

k = 14/4

k = 3.5

Therefore:

C = 3.5m

The unit rate is:

$3.50/kg

For 9 kg:

C = 3.5(9)

C = $31.50

---

Finding a Better Buy

Suppose Store A sells:

5 notebooks for $15.

Unit rate:

15/5 = $3 per notebook

Store B sells:

8 notebooks for $20.

Unit rate:

20/8 = $2.50 per notebook

If each store uses a constant price per notebook, the cost equations are:

Store A:

C = 3n

Store B:

C = 2.5n

Store B has the smaller constant of variation and therefore the lower cost per notebook.

---

Solving a Multi-Step Context Problem

A machine produces 180 components in 12 minutes at a constant rate.

Find the constant of variation.

Let:

C = components produced

t = time in minutes

Use:

k = C/t

k = 180/12

k = 15

Therefore:

C = 15t

How many components are produced in 35 minutes?

C = 15(35)

C = 525

---

Solving for Time

Using:

C = 15t

how long does it take to produce 900 components?

900 = 15t

Divide:

t = 60

Therefore:

60 minutes

are required.

---

Verifying a Proportional Relationship

Suppose someone claims this table represents direct variation:

Check the ratios:

8/2 = 4

16/4 = 4

24/6 = 4

40/10 = 4

The ratio is constant.

Therefore:

k = 4

and the claim is correct.

---

Detecting a Non-Proportional Table

Consider:

The differences are constant:

+3, +3, +3

So it is linear.

But the ratios are:

5/1 = 5

8/2 = 4

11/3 ≈ 3.67

14/4 = 3.5

The ratios are not constant.

Therefore:

The relationship is linear but not direct variation.

---

A Reliable Strategy for Direct Variation Problems

When working with direct variation:

1. Identify the two variables.
2. Determine whether the relationship is proportional.

3. Write the general model:

   y = kx

4. Use known values to calculate:

   k = y/x

5. Include units when interpreting k.
6. Write the complete direct variation equation.
7. Substitute the required input or output.
8. Solve for the unknown.
9. Check that the answer is reasonable.
10. Interpret the result in context.

---

How to Recognize Direct Variation Quickly

Ask:

Equation:
Can it be written as y = kx?

Table:
Is y/x constant?

Graph:
Is it a straight line through (0, 0)?

Situation:
Does one quantity change proportionally with the other, with no fixed starting amount?

If these conditions are satisfied, the relationship represents direct variation.

---

Common Mistakes

Thinking every linear relationship is direct variation

For example:

y = 3x + 5

is linear but not direct variation.

Ignoring the y-intercept

A direct variation must have:

b = 0

Checking only the differences in a table

Constant differences show linearity, but direct variation requires a constant ratio y/x.

Dividing in the wrong order

For:

y = kx

use:

k = y/x

Thinking the constant must be positive

A direct variation can have a negative constant.

Ignoring units

In real-world applications, k often represents a meaningful rate.

Assuming a straight line always represents direct variation

It must also pass through the origin.

Adding a starting value to y = kx

An equation such as:

y = 4x + 10

is no longer direct variation.

Confusing direct variation with inverse variation

In direct variation, increasing x by a factor causes y to increase by the same factor when k is positive. In an inverse relationship, one quantity decreases as the other increases in a different proportional way.

---

Key Terms

Direct variation — relationship in which one variable is a constant multiple of another.

Proportional relationship — relationship with a constant ratio between corresponding quantities.

Constant of variation — constant k in the equation y = kx.

Constant of proportionality — another name for the constant of variation.

Ratio — comparison of two quantities by division.

Unit rate — rate describing the amount of one quantity for one unit of another.

Origin — coordinate point (0, 0).

Slope — rate of change of a linear relationship.

y-intercept — point where a graph crosses the y-axis.

Linear relationship — relationship with a constant rate of change.

Variable — quantity represented using a symbol that can take different values.

Proportion — equation stating that two ratios are equal.

Scaling — increasing or decreasing quantities by a common factor.

---

Key Takeaways

• Direct variation describes a proportional relationship between two variables.
• The general equation is y = kx.
• The value k is the constant of variation.
• The constant can be calculated using k = y/x.
• In a direct variation, the ratio y/x remains constant.
• Direct variation is a special type of linear relationship.
• Every direct variation has a y-intercept of zero.
• The graph of a direct variation is a straight line through the origin.
• The constant of variation is also the slope of the graph.
• Direct variation can have positive, negative, fractional, or decimal constants.
• Multiplying the input by a factor causes the output to be multiplied by the same factor.
• Tables representing direct variation have constant y/x ratios.
• A linear table with a constant difference is not necessarily proportional.
• Equations such as y = mx + b are direct variations only when b = 0.
• The constant of variation often represents a unit rate in real-world situations.
• Direct variation appears in constant-speed motion, hourly pay without a fixed bonus, cost per item, recipe scaling, map scales, and mass-volume relationships at constant density.
• Proportions and direct variation equations are two methods for solving the same type of proportional problem.
• Real-world interpretations should include the units and meaning of the constant.

---

Check Your Understanding

1. Define direct variation in your own words.
2. Identify the constant of variation in y = 7x.
3. Determine whether y = 4x + 3 represents direct variation. Explain your answer.
4. Determine whether the table represents direct variation. If it does, find k and write the equation.

5. Determine whether this table represents direct variation. Explain how you know.

6. If y varies directly with x and y = 28 when x = 4, find the constant of variation and write the equation.
7. Using your equation from Question 6, determine y when x = 15.
8. If y = 6x and y = 102, determine x.
9. Five movie tickets cost $62.50. Assuming the cost varies directly with the number of tickets, find the cost per ticket and determine the cost of 8 tickets.
10. A cyclist travels 42 km in 2.5 hours at constant speed. Write a direct variation equation relating distance and time. Use it to determine the distance travelled in 6 hours.
11. A recipe uses 360 g of flour for 3 batches. Assuming direct variation, find the constant of variation and determine the amount of flour required for 7 batches.
12. A material has a constant density. A 12 cm³ sample has a mass of 96 g. Write an equation relating mass m to volume V and determine the mass of a 35 cm³ sample.
13. Compare y = 2.5x and y = 4x. Identify the constant of variation for each and explain what the larger constant means for positive values of x.
14. A student says that y = 5x + 10 represents direct variation because y increases by 5 whenever x increases by 1. Explain why the student is incorrect and describe what additional condition is required for direct variation.
15. A factory produces 280 components in 16 minutes at a constant rate. Define variables, calculate the constant of variation with units, write a direct variation equation, determine how many components should be produced in 45 minutes, and calculate how long it should take to produce 1050 components.