Algebra

4.5 Direct Variation

Direct Variation Model

The two variables x and y are said to vary directly if their relationship is: y = k x k k is the same as m (slope) 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 = k x 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 .

y = k x 21 = k(7) 7 7 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.

The graph of y = 3x goes through the origin.

All direct variation graphs go through the origin, because when x = 0, y= 0 also.

y 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 = k x 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

84  

kx 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

84  

kx k

(24) 84 2  24 7 

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

84  

kx k

(24) 84  24

y

7 2   7 2

k k

(24)

x

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

84  

kx k

(24) 2.

y

7 2

x

84  24

y

7 2   7 2

k k

(24)

x

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

84  

kx k

(24) 2.

y

7 2

x

84  24

y

7 2   7 2

k k

(24)

x

24

y y

 2 7    35 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.

1.

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.

y



kx

18 

k

1 2

1.

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.

y



kx

18 

k

1 2 (2)18

k

(2)

1.

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.

y



kx

18 

k

1 2 (2)18 36 

k k

(2)

1.

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.

y



kx

18 

k

1 2 (2)18

y

36 

k

 36

x k

(2)

1.

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.

y



kx

2.

y

 36

x

18 

k

1 2 (2)18

y

36 

k

 36

x k

(2)

1.

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.

y



kx

2.

y

 36

x

18 

k

1 2

y y

 36 (5)  180 (2)18

y

36 

k

 36

x k

(2) When x = 5, y = 180

Homework

Pg. 238 #23-31, 38-41