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5-5
Direct Variation
Warm Up
Solve for y.
1. 3 + y = 2x
2. 6x = 3y
Write an equation that describes the
relationship.
3.
Solve for x.
4.
Holt Algebra 1
5.
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Direct Variation
Objective
Identify, write, and graph direct variation.
Holt Algebra 1
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Direct Variation
Vocabulary
direct variation
constant of variation
Holt Algebra 1
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Direct Variation
A recipe for paella calls for 1 cup of rice to make 5
servings. In other words, a chef needs 1 cup of
rice for every 5 servings.
The equation y = 5x describes this relationship. In
this relationship, the number of servings varies
directly with the number of cups of rice.
Holt Algebra 1
5-5
Direct Variation
A direct variation is a special type of linear
relationship that can be written in the form
y = kx, where k is a nonzero constant called
the constant of variation.
Holt Algebra 1
5-5
Direct Variation
Example 1A: Identifying Direct Variations from
Equations
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
y = 3x
Holt Algebra 1
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Direct Variation
Example 1B: Identifying Direct Variations from
Equations
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
3x + y = 8
Holt Algebra 1
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Direct Variation
Example 1C: Identifying Direct Variations from
Equations
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
–4x + 3y = 0
Holt Algebra 1
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Direct Variation
Check It Out! Example 1a
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
3y = 4x + 1
Holt Algebra 1
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Direct Variation
Check It Out! Example 1b
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
3x = –4y
Holt Algebra 1
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Direct Variation
Check It Out! Example 1c
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
y + 3x = 0
Holt Algebra 1
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Direct Variation
What happens if you solve y = kx for k?
y = kx
So, in a direct variation, the ratio is equal to
the constant of variation. Another way to identify
a direct variation is to check whether
is the
same for each ordered pair (except where x = 0).
Holt Algebra 1
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Direct Variation
Example 2A: Identifying Direct Variations from
Ordered Pairs
Tell whether the relationship
is a direct variation. Explain.
Method 1 Write an equation.
Holt Algebra 1
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Direct Variation
Example 2A Continued
Tell whether the relationship
is a direct variation. Explain.
Method 2 Find
Holt Algebra 1
for each ordered pair.
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Direct Variation
Example 2B: Identifying Direct Variations from
Ordered Pairs
Tell whether the relationship
is a direct variation. Explain.
Method 1 Write an equation.
Holt Algebra 1
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Direct Variation
Example 2B Continued
Tell whether the relationship
is a direct variation. Explain.
Method 2 Find
Holt Algebra 1
for each ordered pair.
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Direct Variation
Check It Out! Example 2a
Tell whether the relationship
is a direct variation. Explain.
Method 2 Find
Holt Algebra 1
for each ordered pair.
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Direct Variation
Check It Out! Example 2b
Tell whether the relationship is
a direct variation. Explain.
Method 1 Write an equation.
Holt Algebra 1
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Direct Variation
Check It Out! Example 2c
Tell whether the relationship is
a direct variation. Explain.
Method 2 Find
Holt Algebra 1
for each ordered pair.
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Direct Variation
Example 3: Writing and Solving Direct Variation
Equations
The value of y varies directly with x, and y = 3,
when x = 9. Find y when x = 21.
Method 1 Find the value of k and then write the
equation.
Holt Algebra 1
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Direct Variation
Example 3 Continued
The value of y varies directly with x, and y = 3
when x = 9. Find y when x = 21.
Method 2 Use a proportion.
Holt Algebra 1
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Direct Variation
Check It Out! Example 3
The value of y varies directly with x, and y = 4.5
when x = 0.5. Find y when x = 10.
Method 1 Find the value of k and then write the
equation.
Holt Algebra 1
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Direct Variation
Check It Out! Example 3 Continued
The value of y varies directly with x, and y = 4.5
when x = 0.5. Find y when x = 10.
Method 2 Use a proportion.
Holt Algebra 1
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Direct Variation
Example 4: Graphing Direct Variations
A group of people are tubing down a river at an
average speed of 2 mi/h. Write a direct
variation equation that gives the number of
miles y that the people will float in x hours.
Then graph.
Step 1 Write a direct variation equation.
Holt Algebra 1
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Direct Variation
Example 4 Continued
A group of people are tubing down a river at an
average speed of 2 mi/h. Write a direct
variation equation that gives the number of
miles y that the people will float in x hours.
Then graph.
Step 2 Choose values of x and generate ordered
pairs.
x
Holt Algebra 1
y = 2x
(x, y)
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Direct Variation
Example 4 Continued
A group of people are tubing down a river at an
average speed of 2 mi/h. Write a direct
variation equation that gives the number of
miles y that the people will float in x hours.
Then graph.
Step 3 Graph the points
and connect.
Holt Algebra 1
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Direct Variation
Check It Out! Example 4
The perimeter y of a square varies directly
with its side length x. Write a direct
variation equation for this relationship.
Then graph.
Step 1 Write a direct variation equation.
perimeter
y
Holt Algebra 1
=
4 sides
times
length
=
4
•
x
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Direct Variation
Check It Out! Example 4 Continued
The perimeter y of a square varies directly
with its side length x. Write a direct
variation equation for this relationship.
Then graph.
Step 2 Choose values of x and generate ordered
pairs.
x
Holt Algebra 1
y = 4x
(x, y)
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Direct Variation
Check It Out! Example 4 Continued
The perimeter y of a square varies directly
with its side length x. Write a direct
variation equation for this relationship.
Then graph.
Step 3 Graph the
points and connect.
Holt Algebra 1
5-5
Direct Variation
Lesson Quiz: Part I
Tell whether each equation represents a
direct variation. If so, identify the constant
of variation.
1. 2y = 6x
2. 3x = 4y – 7
Tell whether each relationship is a direct
variation. Explain.
3.
Holt Algebra 1
4.
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Direct Variation
Lesson Quiz: Part II
5. The value of y varies directly with x, and
y = –8 when x = 20. Find y when x = –4.
6. Apples cost $0.80 per pound. The equation
y = 0.8x describes the cost y of x pounds
of apples. Graph this direct variation.
Holt Algebra 1