Direct Variation
Learn to recognize direct variation
and identify the constant of
proportionality.
Spiders
How many legs does a spider have?
 8 legs
 Therefore

2 spiders have a total of 16 legs
 3 spiders have a total of 24 legs
 4 spiders have a total of 32 legs
 And so on. . .


This type of relationship is called . . .
Direct Variation
The relationship between the amount
of spiders and how many legs they
have can be said to vary directly!
 We will be learning about equations,
tables, and graphs of direct variations.
 Sometimes this is called direct
proportion rather than direct variation
but it is the same thing.

Direct Variation
A direct variation relationship can be
represented by a linear equation in
the form y = kx, where k is a positive
number called the constant of
proportionality.
 The constant of proportionality can
sometimes be referred to as the
constant of variation.

y = kx



When two variable quantities have a constant
(unchanged) ratio, their relationship is called a
direct proportion.
We say, “y varies directly as x.”
k is the constant of proportionality which
means it never changes within a problem.
Finding Values Using a
Direct Variation Equation

At a frog jumping contest, Edward’s frog jumped 60
inches. Bella’s frog jumped 72 inches. Jacob’s frog
jumped 6.5 feet. Use the equation y = 12x, where y
is the number of inches and x is the number of feet
to find the missing values of the table.
Edward’s Frog
Bella’s Frog
Jacob’s Frog
Inches (y)
60
72
___
Feet (x)
___
6
6.5
Using Equations and Tables
Edward’s Frog
Bella’s Frog
Jacob’s Frog
Inches (y)
60
72
78
___
Feet (x)
5
___
6
6.5
y  12 x
60  12x
60 12 x

12 12
5 x
Substitute
Solve
Simplify
y  12 x
y  126.5
y  78
Now You Try

Identify the constant of proportionality:
1. y = 15x
15
2. y = .72x
.72
3. y = ¼x
¼
How about solving for k?







When we are given the x and y values, we can solve
for the constant of proportionality.
Example
 If y varies directly with x, and y = 8 when x = 12,
find k and write an equation that expresses this
variation.
Steps:
8 = k × 12 Substitute numbers into y = kx
8/12 = (k × 12)/12 Divide both sides by 12
2/3 = k Simplify
y = 2/3x Plug k back into the equation
Now You Try
If y varies directly as x, and x = 12 when y =
9, what is the equation that describes this
direct variation?
 y = kx
 9 = k × 12
 9/12 = k
¾=k
 y = ¾x

Now You Try
If y varies directly as x, and x = 5 when y
= 10, what is the equation that describes
this direct variation?
 y = kx
 10 = k × 5
 10/5 = k
2=k
 y = 2x

Copy and fill out tables:
y=x
y = 4x
x
y
x
y
1
1
1
4
2
2
2
8
3
3
3
12
4
4
4
16
5
5
5
20
Copy and fill out tables:
y = 10x
y = ½x
x
y
x
y
1
10
1
½
2
20
2
1
3
30
3
3/2
4
40
4
2
5
50
5
5/2
Direct Variation Equation


Sometimes we will have
to put an equation into y =
kx form and solve for y.
Then we will be asked to
identify k, the constant of
proportionality.


1.
2.
3.
Examples:
Tell whether each
equation or
relationship is a direct
variation. If so identify
the constant of
proportionality.
4y = 2x
½y – ¾x = 0
7y – 5 = 3x
More Examples


If y varies directly as x and y = 24 when x = 16, find y
when x = 12.
Solution: Set up a proportion since the ratios of
corresponding values of x to y are always the same.
24 y

16 12
24 12  16  y
288  16 y
288 16 y

16
16
18  y