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11-5
11-5Direct
DirectVariation
Variation
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
Pre-Algebra
11-5 Direct Variation
Warm Up
Use the point-slope form of each
equation to identify a point the line
passes through and the slope of the
line.
1
1. y – 3 = – 1
(x
–
9)
(9, 3), – 7
7
2. y + 2 = 2(x – 5)
3
2
(5, –2), 3
3. y – 9 = –2(x + 4)
(–4, 9), –2
4. y – 5 = – 1 (x + 7)
(–7, 5), – 1
4
4
Pre-Algebra
11-5 Direct Variation
Problem of the Day
Where do the lines defined by the
equations y = -5x + 20 and y = 5x – 20
intersect?
(4, 0)
Pre-Algebra
11-5 Direct Variation
Today’s Learning Goal Assignment
Learn to recognize
direct variation by
graphing tables of data
and checking for
constant ratios.
Pre-Algebra
11-5 Direct Variation
Vocabulary
direct variation
constant of proportionality
Pre-Algebra
11-5 Direct Variation
Pre-Algebra
11-5 Direct Variation
Helpful Hint
The graph of a direct-variation equation is always
linear and always contains the point (0, 0). The
variables x and y either increase together or
decrease together.
Pre-Algebra
11-5 Direct Variation
Additional Example 1A: Determining Whether a Data
Set Varies Directly
Determine whether the data set shows direct
variation.
A.
Pre-Algebra
11-5 Direct Variation
Additional Example 1A Continued
Make a graph that shows the relationship between
Adam’s age and his length.
Pre-Algebra
11-5 Direct Variation
Additional Example 1A Continued
You can also compare ratios to see if a direct
variation occurs.
22 ? 27
3 = 12
81 81 ≠ 264
264
The ratios are not proportional.
The relationship of the data is not a direct
variation.
Pre-Algebra
11-5 Direct Variation
Try This: Example 1A
Determine whether the data set shows direct
variation.
A.
Kyle's Basketball Shots
Distance (ft)
Number of Baskets
Pre-Algebra
20
30
40
5
3
0
11-5 Direct Variation
Try This: Example 1A Continued
Make a graph that shows the relationship between
number of baskets and distance.
Number of Baskets
5
4
3
2
1
20 30 40
Distance (ft)
Pre-Algebra
11-5 Direct Variation
Try This: Example 1A
You can also compare ratios to see if a direct
variation occurs.
5 ? 3
20= 30
60
150
150  60.
The ratios are not proportional.
The relationship of the data is not a direct
variation.
Pre-Algebra
11-5 Direct Variation
Additional Example 1B: Determining Whether a Data
Set Varies Directly
Determine whether the data set shows direct
variation.
B.
Pre-Algebra
11-5 Direct Variation
Additional Example 1B Continued
Make a graph that shows the relationship between
the number of minutes and the distance the train
travels.
Plot the points.
The points lie in
a straight line.
(0, 0) is included.
Pre-Algebra
11-5 Direct Variation
Additional Example 1B Continued
You can also compare ratios to see if a direct
variation occurs.
25 = 50= 75 = 100
10 20 30 40
Compare ratios.
The ratios are proportional. The relationship is
a direct variation.
Pre-Algebra
11-5 Direct Variation
Try This: Example 1B
Determine whether the data set shows direct
variation.
B.
Ounces in a Cup
Ounces (oz)
8
16
24
32
Cup (c)
1
2
3
4
Pre-Algebra
11-5 Direct Variation
Try This: Example 1B Continued
Number of Cups
Make a graph that shows the relationship between
ounces and cups.
Plot the points.
4
3
The points lie in
a straight line.
2
(0, 0) is included.
1
8
16 24 32
Number of Ounces
Pre-Algebra
11-5 Direct Variation
Try This: Example 1B Continued
You can also compare ratios to see if a direct
variation occurs.
1= 2 = 3 = 4
8 16 24 32
Compare ratios.
The ratios are proportional. The relationship is
a direct variation.
Pre-Algebra
11-5 Direct Variation
Additional Example 2A: Finding Equations of Direct
Variation
Find each equation of direct variation, given
that y varies directly with x.
A. y is 54 when x is 6
y = kx
54 = k  6
y varies directly with x.
Substitute for x and y.
9=k
Solve for k.
y = 9x
Substitute 9 for k in the original
equation.
Pre-Algebra
11-5 Direct Variation
Try This: Example 2A
Find each equation of direct variation, given
that y varies directly with x.
A. y is 24 when x is 4
y = kx
24 = k

y varies directly with x.
4
Substitute for x and y.
6=k
Solve for k.
y = 6x
Substitute 9 for k in the original
equation.
Pre-Algebra
11-5 Direct Variation
Additional Example 2B: Finding Equations of Direct
Variation
B. x is 12 when y is 15
y = kx
15 = k

5=k
4
y =5
4k
Pre-Algebra
y varies directly with x.
12
Substitute for x and y.
Solve for k.
5
Substitute 4 for k in the original
equation.
11-5 Direct Variation
Try This: Example 2B
B. x is 28 when y is 14
y = kx
14 = k

1=k
2
y =1
2k
Pre-Algebra
y varies directly with x.
28
Substitute for x and y.
Solve for k.
1
Substitute 2 for k in the original
equation.
11-5 Direct Variation
Additional Example 2C: Finding Equations of Direct
Variation
C. y is 8 when x is 5
y = kx
8=k

8=k
5
y =8
5k
Pre-Algebra
y varies directly with x.
5
Substitute for x and y.
Solve for k.
8
Substitute 5 for k in the original
equation.
11-5 Direct Variation
Try This: Example 2C
C. y is 7 when x is 3
y = kx
7=k

7=k
3
y =7
3k
Pre-Algebra
y varies directly with x.
3
Substitute for x and y.
Solve for k.
7
Substitute 3 for k in the original
equation.
11-5 Direct Variation
Additional Example 3: Money Application
Mrs. Perez has $4000 in a CD and $4000 in a
money market account. The amount of
interest she has earned since the beginning of
the year is organized in the following table.
Determine whether there is a direct variation
between either of the data sets and time. If
so, find the equation of direct variation.
Pre-Algebra
11-5 Direct Variation
Additional Example 3 Continued
A. interest from CD and time
interest from CD 17
interest from CD 34
=
= 2 = 17
time
1
time
The second and third pairs of data result in a common
ratio. In fact, all of the nonzero interest from CD to
time ratios are equivalent to 17.
interest from CD = 17 = 34 = 51 = 68 = 17
time
1
2
3
4
The variables are related by a constant ratio of 17 to
1, and (0, 0) is included. The equation of direct
variation is y = 17x, where x is the time, y is the
interest from the CD, and 17 is the constant of
proportionality.
Pre-Algebra
11-5 Direct Variation
Additional Example 3 Continued
B. interest from money market and time
interest from money market = 19 = 19
time
1
interest from money market = 37 =18.5
time
2
19 ≠ 18.5
If any of the ratios are not equal, then there
is no direct variation. It is not necessary to
compute additional ratios or to determine
whether (0, 0) is included.
Pre-Algebra
11-5 Direct Variation
Try This: Example 3
Mr. Ortega has $2000 in a CD and $2000 in a
money market account. The amount of
interest he has earned since the beginning of
the year is organized in the following table.
Determine whether there is a direct variation
between either of the data sets and time. If
so, find the equation of direct variation.
Pre-Algebra
11-5 Direct Variation
Try This: Example 3 Continued
Time (mo)
0
Interest
from CD ($)
0
Interest from
Money Market ($)
0
1
2
3
4
12
30
40
50
15
40
45
50
Pre-Algebra
11-5 Direct Variation
Try This: Example 3 Continued
A. interest from CD and time
interest from CD = 12
time
1
interest from CD = 30 = 15
time
2
The second and third pairs of data do not result in a
common ratio.
If any of the ratios are not equal, then there
is no direct variation. It is not necessary to
compute additional ratios or to determine
whether (0, 0) is included.
Pre-Algebra
11-5 Direct Variation
Try This: Example 3 Continued
B. interest from money market and time
interest from money market = 15 = 15
time
1
interest from money market = 40 =20
time
2
15 ≠ 20
If any of the ratios are not equal, then there
is no direct variation. It is not necessary to
compute additional ratios or to determine
whether (0, 0) is included.
Pre-Algebra
11-5 Direct Variation
Lesson Quiz: Part 1
Find each equation of direct variation, given
that y varies directly with x.
1. y is 78 when x is 3.
2. x is 45 when y is 5.
3. y is 6 when x is 5.
Pre-Algebra
y = 26x
y =1 x
9
y =6 x
5
11-5 Direct Variation
Lesson Quiz: Part 2
4. The table shows the amount of money Bob
makes for different amounts of time he works.
Determine whether there is a direct variation
between the two sets of data. If so, find the
equation of direct variation.
direct variation; y = 12x
Pre-Algebra