10.2 Inverse and Joint Variation(1)
Objective: To create and solve direct and inverse variation equations.
DO NOW: Solve for x
3 7
1.) 
x 49
2.) 4 x  5  6
3.)
6 x

5 8
What is direct/joint variation?
Verbal Example: The more hours Geoff works, the more money he makes.
Equation:
Graph:
Direct/Joint Variation
y  3x
x
2
As x & z increase,
15
y ___________________
27
Equation:
y = kx
y = kxz
7
The graph of (x,y) pairs
form a ______________.
Solving Direct Variation Problems
Examples: If y varies directly as x and y = 8 when x = 10, what is y when x = 20?
Method 1:
Use the
equation
y  kx .
Step 1: Find k, the constant of variation.
y  kx
( y varies directly with x)
8  k10
 y is 8 when x is 10 
Step 2: Use k to find the missing
variable
y  .8 x
y  .8(20)
( x  20)
y  16
.8  k
(solve for k )
For all direct variation problems, a proportion can be used instead of the equation. If y  kx , then
y
y
y
 k for all pairs of x and y. In other words, 1  2 for all pairs of x and y.
x
x1 x2
Method 2:
Use the
proportion
y1 y2

x1 x2
Step 1: Plug the given
values into the
proportion.
8
y

10 20
Step 2:
Solve for the missing variable.
Multiply both sides by 20:
8
y
20   20 
10
20
Simplify:
16  y
Example 1:
1.) If y varies directly as x and y = 15 when x = 6, what is y when x = 2?
2.) If y varies directly as x and y = 60 when x = 55, what is x when y = 12?
You Try:
3.) If y varies directly as x and y = 100 when x =14, what is y when x = 56?
4.) If y varies directly as x and y = 33 when x = 55, what is x when y = 45?
What is inverse variation?
Example: The more Lisa exercises, the less body fat she has.
Equaton:
Graph:
Inverse Variation
2
y
x
x
As x increases,
1
y _________________
1
6
10
Equation:
xy = k or y = k/x
The graph of (x,y) pairs
form a ______________.
Solving Inverse Variation Problems
Example: If y varies inversely as x and y = 8 when x = 10, what is y when x = 20?
Method: Since
xy  k ,
the products
are always
equal;
x1 y1  x2 y2
Step 1: Plug the given values into the product.
x1 y1  x2 y2
(8)(10)  (20)( y)
80  20 y
Step 2: Solve for the missing
variable.
80  20 y
80 20 y

20 20
4 y
Example 2:
1.) If y varies inversely as x and y = 20 when x = 6, what is y when x = 2?
2.) If y varies inversely as x and y = 60 when x = 55, what is x when y = 12?
You Try:
3.) If y varies inversely as x and y = 100 when x =14, what is y when x = 56?
4.) If y varies inversely as x and y = 33 when x = 15, what is x when y = 45?
Example 3: For each of the following, tell whether the relationship is a direct relationship or an
inverse relationship
1) If y is the true distance between two cities and x is the distance on a given map.
2) If y is the radius of a circle and x is the circumference of the circle.
3) If y is time it takes to drive to Newark and x is your average speed on the drive.
Example 4: Determine if the following are inverse, direct or joint variation.
a)
x
=6
y
b) 2y =
1
x
c)
xyz
=2
6
Example 5: Write an equation using the given information.
a) Inverse x = -2, y = 3
b) Joint: x = 2, y = 7, z = ½