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Exercise
Solve.
5 = 3x
5
x=
3
Exercise
Solve.
1
4= k
3
k = 12
Exercise
Solve.
3 = y
5 25
y = 15
Exercise
Solve.
3 = 25
5
x
2
x = 41
3
Exercise
If 4 = 6k, what is the value
of 3k?
2
Direct Variation
A direct variation is
formed by the variables
x and y if the ratio y : x
always equals a
constant k, where k is a
positive number.
Directly Proportional
Variables are directly
proportional when y is
said to vary directly
with x.
Constant of
Proportionality
The constant k is the
constant of variation, or
the constant of
proportionality.
x hours y miles
1
2
3
4
30
60
90
120
y
x
Example 1
Does y vary directly with x in
the following table? If so, find
the constant of variation and
write an equation for the
direct variation.
x 1 3 5 7
y 3 9 15 21
x 1 3 5 7
y 3 9 15 21
y
3
=
=
3
x
1
y = 9 =3
x
3
y = 15 = 3
x
5
y = 21 = 3
x
7
y = 3 =k
x
y = kx
y = 3x
Constant of Variation
The constant of variation is
the steady rate of change.
The constant k is the
constant of variation, or the
constant of proportionality.
Example 2
Indicate which equations
represent a direct variation. If
an equation describes a
direct variation, give the
constant of variation.
f(x) = 2.2x direct variation;
k = 2.2
y = 4x − 1 This is not a direct
variation; the variable
must be a multiple of x.
d = 45t This is a direct
variation; k = 45.
y = −2x This is not a direct
variation; the coefficient
of x must be positive.
y = mx + b
y = kx
Example 3
Graph the direct variation
y = 4x.
x y
−1 −4
0
0
1
4
y
x
Example 4
Find k if y varies directly with
1
x and y = 12 when x = 2 .
Write an equation for the
direct variation.
y = kx
k = 24
1
12 = k( 2 )
y = 24x
2(12) =
1
k( 2 )(2)
Example 5
If y varies directly with x and
y = 6 when x = 2, find y when
2
x= 3.
y = kx
6 = k(2)
3=k
y = 3x
2
y = 3( 3 )
y=2
Example
Find k if y varies directly with
x and y = 14 when x = 4.
k = 3.5
Example
Find k if y varies directly with
x and y = 15 when x = 2.
k = 7.5
Example
If y varies directly with x and
y = 7 when x = 1, find y when
x = 6.
y = 42
Example
If y varies directly with x and
y = 27 when x = 15, find y
when x = 6.
54
y=
5
Example
Indicate which equations
represent a direct variation. If
an equation describes a
direct variation, give the
constant of variation. If not,
explain.
y = 4x Yes. k = 4.
y = 3x + 5 No. The y-intercept
is not zero.
y = −4x
No. The slope is
not positive.