Directvariation

Direct Variation
What is it and how do I know when I see it?

Definition:
Y varies directly as x means that y = kx
where k is the constant of variation.
y
Another way of writing this is k =
x

In other words:
* the constant of variation (k) in a direct
variation is the constant (unchanged) ratio of two
variable quantities.

Examples of Direct Variation:
X
6
7
8

Y
12
14
16

Note: X increases,
6,7,8
And Y increases.
12, 14, 16

What is the constant of variation of the table above?
y
Since y = kx we can say k =
Therefore:
x
12/6=k or k = 2
14/7=k or k = 2
16/8=k or k =2

Note k stays constant.

y = 2x is the
equation!

Note: X decreases,

X
10
5
3

Y
30
15
9

10, 5, 3
And Y decreases.
30, 15, 9


y
Therefore:
x
30/10=k or k = 3
15/5=k or k = 3
9/3=k or k =3

Note k stays constant.

y = 3x is the
equation!

Note: X decreases,

X
-4
-16
-40

Y
-1
-4
-10

-4, -16, -40
And Y decreases.
-1,-4,-10

y
Therefore:
x
y = ¼ x is the
-1/-4=k or k = ¼
-4/-16=k or k = ¼
equation!
-10/-40=k or k = ¼ Note k stays constant.

What is the constant of variation for the
following direct variation?
1.
2.
3.
4.

X
4
8
-6
3

2
-2
-½
½
Answer
Now

Y
-8
-16
12
-6

Is this a direct variation? If yes, give the
constant of variation (k) and the equation.

X
4
8
12
18

Y
6
12
18
27

Yes!
k = 6/4 or 3/2
Equation?
y = 3/2 x


X
10
6
4
2

Y
25
15
10
5

Yes!
k = 25/10 or 5/2
k = 10/4 or 5/2
Equation?
y = 5/2 x


X
15
3
1
2

Y
5
26
75
150

No!
The k values are
different!

Which of the following is a direct variation?

0%

0%

0%

0%
D

4.

Answer
Now

C

3.

B

2.

A
B
C
D

A

1.

Which is the equation that describes the
following table of values?

0%

0%

0%
20
0
=
xy

=

=

½

2x

x

0%

y

Answer
Now

y

4.

Y
5
1
6
10

-2
x

3.

X
10
2
12
20

=

2.

y = -2x
y = 2x
y= ½x
xy = 200

y

1.

Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 28 when
x=7, Find x when y = 52.
HOW???
2 step process

k = y/x or k = 28/7 = 4
k=4

Y

7

28

?

1. Find the constant variation

X

52

2. Use y = kx. Find the unknown (x).
52= 4x or 52/4 = x
x= 13

Therefore:
X =13 when Y=52

Given that y varies directly with x, and y = 3 when x=9,
Find y when x = 40.5.

HOW???

2 step process
1. Find the constant variation.
k = y/x or k = 3/9 = 1/3
K = 1/3

X

Y

9

3

40.5

?

y= (1/3)40.5
y= 13.5

Therefore:
X =40.5 when
Y=13.5

Given that y varies directly with x, and y = 6 when x=-5,
Find y when x = -8.

HOW???

2 step process
1. Find the constant variation.
k = y/x or k = 6/-5 = -1.2
k = -1.2

X

Y

-5

6

-8

?

y= -1.2(-8)

Therefore:

x= 9.6

X =-8 when Y=9.6

Using Direct Variation to solve word problems
Problem:
A car uses 8 gallons of
gasoline to travel 290
miles. How much
gasoline will the car use
to travel 400 miles?
Step Two: Find the constant
variation and equation:
k = y/x or k = 290/8 or 36.25
y = 36.25 x

Step One: Find points in table

X (gas) Y (miles)
8
290
?
400

Step Three: Use the equation
to find the unknown.
400 =36.25x
400 =36.25x
36.25 36.25
or x = 11.03

Using Direct Variation to solve word problems
Problem:
Julio wages vary
directly as the number
of hours that he works.
If his wages for 5 hours
are $29.75, how much
will they be for 30 hours
Step Two: Find the constant
variation.
k = y/x or k = 29.75/5 = 5.95

Step One: Find points in table.

X(hours) Y(wages)
5
29.75
30
?
Step Three: Use the equation
to find the unknown. y=kx
y=5.95(30) or Y=178.50

Direct Variation and its graph

With direction variation the equation
is y = kx

Tell if the following graph is a Direct Variation or not.

No

No

Yes

No

Tell if the following graph is a Direct Variation or not.

No

Yes

Yes

No

Directvariation

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