Solve for a.

Warm-Up Exercises

1. 2a10 =

2. 165a = –

3. Write an equation of the line that passes through the points and . ( )0, 0 ( )4, 8

ANSWER 5

ANSWER5

16–

ANSWER 2xy =

4. A flower is 4.5 centimeters wide and has a leaf 7.2 centimeters long. What is the ratio of flower width to leaf length?

ANSWER 5 to 8

Example 1 Graph a Direct Variation Equation

Graph .4

7y x=

Find a second point by substituting a convenient value for x.

4

7y x= Write equation.

4

7y 4= ( ) Substitute 4 for x.

SOLUTION

Plot a point at the origin.

Example 1 Graph a Direct Variation Equation

7y = Simplify.

A second point is .4, 7( )

Plot the second point. Then draw a line through the two points as shown.

Checkpoint

Graph the equation.

ANSWER

xy = 22. ANSWER

Graph a Direct Variation Equation

x–1. y =

Checkpoint

Graph the equation.

ANSWER

ANSWER

Graph a Direct Variation Equation

4xy = –3.

4.3

2y x=

Example 2 Write a Direct Variation Equation

Recipes To make mango salsa, replace the tomatoes in the recipe above with mangoes, using 3 mangoes for 2 recipes. Write an equation relating the number of servings y to the number of mangoes x given so that y varies directly with x. How many servings of salsa can you make if you have 5 mangoes?

Example 2 Write a Direct Variation Equation

In 2 recipes there are servings, so 3 mangoes make 8 servings. This means when . Write an equation using these values of y and x to solve for k.

SOLUTION

( )42 8=8y = 3x =

kxy = Write a direct variation equation.

8 = Substitute 8 for y and 3 for x.( )3k

= Simplify.k3

8

An equation is . To find how many servings of

salsa you can make using 5 mangoes, substitute 5 for x.

= x3

8y

Example 2 Write a Direct Variation Equation

= x3

8y =

3

8 ( )53

40= =

3

113

ANSWER

Five mangoes will make about 13 servings.

Checkpoint

The variables x and y vary directly. Write an equation that relates x and y.

5. x = 3, y = 9– ANSWER y = 3x–

6. x = 10, y = 2 ANSWER = x5

1y

Write a Direct Variation Equation

Example 3 Identify Direct Variation

Tell whether the data, the numbers of words y in the first x lines of the Declaration of Independence, show direct variation. If so, write an equation relating x and y.

SOLUTION

Find the ratio of y to x for each data pair.

30.6=5

15330.5=

10

30530.1=

20

60230.27

15

454≈ 30.28

25

757≈

Words, y 305

15

454

20

602

25

757

Lines, x 5 10

153

Example 3 Identify Direct Variation

The ratio of y to x in each pair is nearly the same, so the data show direct variation. Substituting 30.3 for k in y kx gives the direct variation equation y 30.3x.= =

Checkpoint

7. Tell whether the data show direct variation. If so, write an equation relating x and y.

( ),1, 2 ( ),2, 4 ( ),3, 8 ( ),4, 14 ( )5, 22

ANSWER not direct variation

Identify Direct Variation