Algebra
4.5
Direct Variation
Direct Variation Model
The two variables x and y are said to
vary directly if their relationship is:
y = kx
k is the same as m (slope)
k is called the constant of variation
The price of hot dogs varies directly with
the number of hotdogs you buy
You buy hotdogs.
x represents the number of hotdogs you buy.
y represents the price you pay.
y = kx
y = kx
21 = k(7)
7
7
Let’s figure out k, the price per hotdog.
Suppose that when you buy 7 hotdogs, it costs $21.
Plug that information into the model to solve for k.
Now divide both sides by 7
to solve for k.
k=3
The price per hotdog is $3.
y = 3x
You could use this model to find the price (y)
for any number of hotdogs (x) you buy.
y
The graph of y = 3x
goes through the origin.
All direct variation
graphs go through the
origin, because when x =
0, y= 0 also.
x
y (price)
y = 3x
.
.
.
.
(3,9) When you buy 3 hotdogs, you pay $9
(2,6) When you buy 2 hotdogs, you pay $6
(1,3) When you buy 1 hotdog, you pay $3
x (number of hotdogs)
(0,0) When you buy 0 hotdogs, you pay $0
Finding the Constant of Variation (k)
STEPS
1. Plug in the known values for x and y
into the model: y = kx
2. Solve for k
3. Now write the model y = kx and
replace k with the number
4. Use the model to find y for other
values of x if needed
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1. y  kx
84  k (24)
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1. y  kx
84  k (24)
84

24
7
2
 k
k (24)
24
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1. y  kx
84  k (24)
84

24
7
k (24)
24
 k
2
y 
7
2
x
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1. y  kx
2.
84  k (24)
84

24
7
k (24)
24
 k
2
y 
7
2
x
y 
7
2
x
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1. y  kx
2.
y 

24
7
k (24)
24
 k
y 
2
y 
7
2
10 
y  35
2
7
x
2
84  k (24)
84
7
x
When x = 10, y = 35
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
 1 
18  k 

 2 
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
 1 
18  k 

 2 
1
(2)18  k   (2)
2
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
 1 
18  k 

 2 
1
(2)18  k   (2)
2
36  k
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
 1 
18  k 

 2 
1
(2)18  k   (2)
2
36  k
y  36 x
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
 1 
18  k 

 2 
1
(2)18  k   (2)
2
36  k
y  36 x
2.
y  36 x
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
 1 
18  k 

 2 
1
(2)18  k   (2)
2
36  k
y  36 x
2.
y  36 x
y  36 (5)
y  180
When x = 5, y = 180
Homework
Pg. 238 #23-31, 38-41