Direct Variation and Proportion

Objective: Write and apply direct variation and proportion problems

Direct Variation

• The variable y varies directly as x if there is a nonzero constant k such that y = kx. The equation y = kx is called a direct variation equation and the number k is called the constant of variation.

Example 1

• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.

Example 1

• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.

• Use the direct variation equation• Substitute -24 for y and 4 for x 424

k

kxy

Example 1

• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.

• Use the direct variation equation• Substitute -24 for y and 4 for x• Solve for k.

k

k

k

kxy

6

424

424

Example 1

• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.

• Use the direct variation equation• Substitute -24 for y and 4 for x• Solve for k.

• The direct variation equation is

k

k

k

kxy

6

424

424

xy 6

Try This

• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = 21 when x = 7.

Try This

• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = 21 when x = 7.

• Use the direct variation equation• Substitute 21 for y and 7 for x• Solve for k.

• The direct variation equation is

k

k

k

kxy

3

721

721

xy 3

Example 2

• Each day John rides his bike for exercise. When traveling at a constant rate, he rides 4 miles in about 20 minutes. At this rate, how long would it take John to travel 7 miles?

Example 2

• Each day John rides his bike for exercise. When traveling at a constant rate, he rides 4 miles in about 20 minutes. At this rate, how long would it take John to travel 7 miles?

Example 2

Proportions

• A ratio is a comparison of two numbers by division.• A proportion is a statement that two ratios are equal.

Proportions

• A ratio is a comparison of two numbers by division.• A proportion is a statement that two ratios are equal.

• A proportion of the form

can be rearranged like this:

d

c

b

a

bcad

bdd

c

b

abd

Cross-Product Property

• The result of what we just did is called the cross-product property of proportions. This is also called the means-extremes product property.

:0 and 0For db

bcadd

c

b

a then , If

Physics Application

Example 3

Example 3

Example 3

Example 4

Example 4

Class work

• In pairs/groups• Page 33• 8-13

Homework

• Pages 33-34• 15-25 odd• 37-51 odd