Direct variation
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Direct Variation. 11-5. Pre-Algebra. 1. 1. 2. 1. 1. 2. 3. 7. 4. 4. 3. 7. (9, 3), –. (5, – 2), . (–7, 5), –. 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. y – 3 = – ( x – 9)

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Direct Variation

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Direct variation

Direct Variation

11-5

Pre-Algebra


Warm up

1

1

2

1

1

2

3

7

4

4

3

7

(9, 3), –

(5, –2),

(–7, 5), –

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.y – 3 = – (x – 9)

2.y + 2 = (x – 5)

3.y – 9 = –2(x + 4)

4.y – 5 = – (x + 7)

(–4, 9), –2


Direct variation

Learn to recognize direct variation by graphing tables of data and checking for constant ratios.


Vocabulary

Vocabulary

direct variation

constant of proportionality


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.


Example determining whether a data set varies directly

Example: Determining Whether a Data Set Varies Directly

Determine whether the data set shows direct variation.

A.


Example continued

Example Continued

Make a graph that shows the relationship between Adam’s age and his length.


Example continued1

27

22

12

3

?

=

Example Continued

You can also compare ratios to see if a direct variation occurs.

81

81 ≠ 264

The ratios are not proportional.

264

The relationship of the data is not a direct variation.


Example determining whether a data set varies directly1

Example: Determining Whether a Data Set Varies Directly

Determine whether the data set shows direct variation.

B.


Example continued2

Example 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.


Example continued3

25

10

75

100

50

30

40

20

Example Continued

You can also compare ratios to see if a direct variation occurs.

Compare ratios.

=

=

=

The ratios are proportional. The relationship is a direct variation.


Try this

Try This

Determine whether the data set shows direct variation.

A.


Try this continued

Try This Continued

Make a graph that shows the relationship between number of baskets and distance.

5

4

3

Number of Baskets

2

1

20

30

40

Distance (ft)


Try this1

3

5

30

20

?

=

Try This

You can also compare ratios to see if a direct variation occurs.

60

150  60.

The ratios are not proportional.

150

The relationship of the data is not a direct variation.


Try this2

Try This

Determine whether the data set shows direct variation.

B.


Try this continued1

4

3

Number of Cups

2

1

32

8

16

24

Number of Ounces

Try This Continued

Make a graph that shows the relationship between ounces and cups.

Plot the points.

The points lie in a straight line.

(0, 0) is included.


Try this continued2

1

=

=

=

8

3

4

2

24

32

16

Try This Continued

You can also compare ratios to see if a direct variation occurs.

Compare ratios.

The ratios are proportional. The relationship is a direct variation.


Example finding equations of direct variation

Example: 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

y varies directly with x.

54 = k6

Substitute for x and y.

9 = k

Solve for k.

Substitute 9 for k in the original equation.

y = 9x


Example finding equations of direct variation1

= k

y = k

Substitute for k in the original equation.

5

5

5

4

4

4

Example: Finding Equations of Direct Variation

B. x is 12 when y is 15

y = kx

y varies directly with x.

15 = k12

Substitute for x and y.

Solve for k.


Example finding equations of direct variation2

= k

y = k

Substitute for k in the original equation.

8

8

8

5

5

5

Example: Finding Equations of Direct Variation

C. y is 8 when x is 5

y = kx

y varies directly with x.

8 = k5

Substitute for x and y.

Solve for k.


Try this3

Try This

Find each equation of direct variation, given that y varies directly with x.

A. y is 24 when x is 4

y = kx

y varies directly with x.

24 = k4

Substitute for x and y.

6 = k

Solve for k.

Substitute 9 for k in the original equation.

y = 6x


Try this4

= k

y = k

Substitute for k in the original equation.

1

1

1

2

2

2

Try This

B. x is 28 when y is 14

y = kx

y varies directly with x.

14 = k28

Substitute for x and y.

Solve for k.


Try this5

= k

y = k

Substitute for k in the original equation.

7

7

7

3

3

3

Try This

C. y is 7 when x is 3

y = kx

y varies directly with x.

7 = k3

Substitute for x and y.

Solve for k.


Example money application

Example: 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.


Example continued4

= = = 17

68

34

51

2

4

3

interest from CD

interest from CD

interest from CD

17

17

34

= = 17

=

= =

time

time

time

2

1

1

Example Continued

A. interest from CD and 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.

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.


Example continued5

19

37

1

2

interest from money market

interest from money market

= = 19

= =18.5

time

time

Example Continued

B. interest from money market and time

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.


Try this6

Try This

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.


Try this continued3

Try This Continued


Try this continued4

interest from CD

interest from CD

12

30

=

= = 15

time

time

1

2

Try This Continued

A. interest from CD and time

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.


Try this continued5

15

40

1

2

interest from money market

interest from money market

= = 15

= =20

time

time

Try This Continued

B. interest from money market and time

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.


Lesson quiz part 1

y = x

y = x

1

6

9

5

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.

y = 26x


Lesson quiz part 2

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


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