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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), –

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

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

direct variation

constant of proportionality

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.

Determine whether the data set shows direct variation.

A.

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

27

22

12

3

?

=

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.

Determine whether the data set shows direct variation.

B.

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.

25

10

75

100

50

30

40

20

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

Compare ratios.

=

=

=

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

Determine whether the data set shows direct variation.

A.

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)

3

5

30

20

?

=

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.

Determine whether the data set shows direct variation.

B.

4

3

Number of Cups

2

1

32

8

16

24

Number of Ounces

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.

1

=

=

=

8

3

4

2

24

32

16

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

Compare ratios.

The ratios are proportional. The relationship is a 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 = k6

Substitute for x and y.

9 = k

Solve for k.

Substitute 9 for k in the original equation.

y = 9x

= k

y = k

Substitute for k in the original equation.

5

5

5

4

4

4

B. x is 12 when y is 15

y = kx

y varies directly with x.

15 = k12

Substitute for x and y.

Solve for k.

= k

y = k

Substitute for k in the original equation.

8

8

8

5

5

5

C. y is 8 when x is 5

y = kx

y varies directly with x.

8 = k5

Substitute for x and y.

Solve for k.

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 = k4

Substitute for x and y.

6 = k

Solve for k.

Substitute 9 for k in the original equation.

y = 6x

= k

y = k

Substitute for k in the original equation.

1

1

1

2

2

2

B. x is 28 when y is 14

y = kx

y varies directly with x.

14 = k28

Substitute for x and y.

Solve for k.

= k

y = k

Substitute for k in the original equation.

7

7

7

3

3

3

C. y is 7 when x is 3

y = kx

y varies directly with x.

7 = k3

Substitute for x and y.

Solve for k.

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.

= = = 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

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.

19

37

1

2

interest from money market

interest from money market

= = 19

= =18.5

time

time

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.

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.

interest from CD

interest from CD

12

30

=

= = 15

time

time

1

2

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.

15

40

1

2

interest from money market

interest from money market

= = 15

= =20

time

time

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.

y = x

y = x

1

6

9

5

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

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