Direct, inverse, and joint variation are types of proportional relationships between two or more variables. Direct variation means one variable (y) varies directly with another (x) in the form of y=kx, where k is the constant of variation. Inverse variation means y varies inversely with x in the form of y=k/x. Joint variation means y varies jointly with two or more variables, like x and z, in the form of y=kxz. Examples are provided to demonstrate calculating values for each type of variation relationship given information about the variables.

1. Direct, Inverse and Joint
Variation

2. Direct Variation
• y varies directly as x if there is some nonzero
constant k such that y = kx. k is called the
constant of variation

3. Example
• If y varies directly as x and y = 9 when x is -15,
find y when x = 21.
kx
y 
)
15
(
9 
 k
k


5
3
x
y
5
3


)
21
(
5
3


y
5
63


y

4. Inverse Variation
• y varies inversely as x if there is some nonzero
constant k such that xy = k or y = k/x

5. Example
• If y varies inversely as x and y = 4, when x = 12,
find y when x = 5.
x
k
y 
12
4
k

k

48
x
y
48

5
48

y

6. Joint Variation
• y varies jointly as x and z if there is some
number k such that y = kxz, where
and
0

x
0

z

7. Example
• The area A of a trapezoid varies jointly as the
height h and the sum of its bases b1 and b2. Find
the equation of joint variation if A = 48 in^3, h =
8 in., b1 = 5 in. and b2 = 7 in.
kxz
y 
2
1

k

8. Practice
• If y varies jointly as x and z and y = 45 when x
= 9 and z = 15, find y when x = 25 and z = 12
100

y

9. Practice
• If y varies inversely as x and y = ¼ when
x = 24, find y when x = ¾
y = 8

10. YU TRY!
• If m varies directly as w and m = -15
when w = 2.5, find m when w = 12.5
y = ?
Senteo Question
To set the properties right click and select
A: 75 B:-25 C: -75 D: 25

11. Example
• The power P in watts of an electrical
circuit varies jointly as the resistance R
and the square of the current I. For a 600-
watt microwave oven that draws a current
of 5.0 amperes, the resistance is 24 ohms.
• What is the resistance of a 200-watt
refrigerator that draws a current of 1.7
amperes?

12. 2
kRI
P 
2
)
5
)(
24
(
600 k

600
600 k

k

1
2
RI
P 
2
)
7
.
1
(
200 R

89
.
2
200 R

R

69

13. YU Practice
• The LEM (Lunar Exploration Module) used by
astronauts to explore the moon’s surface during
the Apollo space missions weighs about 30,000
pounds on Earth. On the moon there is less
gravity so it weighs less, meaning that less fuel
is needed to lift off from the moon’s surface. The
force of gravity on Earth is about 6 times as
much as that on the moon. How much does the
LEM weigh on the moon?
Senteo Question
Enter the number pounds in your clicker!

14. Answer
• y =kx
• 30,000 = 6x
• x = 5,000 lbs