Representing proportional relationships with equations

What we’ve learned…..

• Proportional relationships have a constant ratio, or unit rate.

• The graph of a proportional relationship is a straight line that passes through the origin.

New vocabulary…..• A proportional relationship can be called a

direct variation.

• The constant ratio, or unit rate, can also be called the constant of proportionality.

• Direct variations can be represented by the equation y = kx where k is the constant of proportionality.

Writing a direct variation equation from the graph of a proportional relationship

Ex. 1) The number of chaperones needed varies directly with the number of students going on a trip.

A) Find the constant of proportionality k, or the unit rate, of this relationship.

B) Write an equation to represent this relationship.

chaperones

students

1

6

1

6

2

12

1

6

3

18

1

6

4

24

chaperone 1

students 6k

y = kxNo. of students No. of chaperones

y = 6x

(1, 6)

(2, 12)

(3, 18)

(4, 24)

Writing a direct variation equation from a table of values.

Ex. 2) In Mr. Miller’s new car, the number of miles driven varies directly with the gallons of gas used.

A) Find the constant of proportionality k, or the unit rate, of this relationship.

B) Write an equation to represent this relationship.

gallons

miles

x

y

1

30

6

180 1

30

8

240

1

30

10

300

1

30

12

360

gallon 1

miles 30k

y = kxNo. of miles

Gallons of gas used

y = 30x

Writing a direct variation equation from a verbal statement.

Ex. 3) If y varies directly with x, write the equation for the direct variation if y = 6 when x = 2.

A) Find the constant of proportionality k, or the unit rate, of this relationship.

B) Write an equation to represent this relationship.

x

yk

y = kx

y = 3x

32

6k

C) What is the value of y when x=7? y = 3xy = 3(7)y = 21 when x = 7