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Solve.Solve.ExerciseExercise
5 = 3x5 = 3x
x =x = 5353
4 = k4 = k1313
Solve.Solve.
k = 12k = 12
ExerciseExercise
==y
25y
25
Solve.Solve.
y = 15y = 15
3535
ExerciseExercise
== 25x
25x
Solve.Solve.3535
x = 41x = 41 2323
ExerciseExercise
If 4 = 6k, what is the value of 3k?If 4 = 6k, what is the value of 3k?
22
ExerciseExercise
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.
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.
Direct VariationDirect Variation
Variables are directly proportional when y is said to vary directly with x.
Variables are directly proportional when y is said to vary directly with x.
Directly ProportionalDirectly Proportional
The constant k is the constant of variation, or the constant of proportionality.
The constant k is the constant of variation, or the constant of proportionality.
Constant of ProportionalityConstant of Proportionality
3 904 120
2 601 30
x hours y miles yx
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.
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.
xy
13
39
515
721
Example 1Example 1
xy
13
39
515
721
y = 3xy = 3x
==3131
yxyx == 33
== 9393
yxyx
== 33
== 155
155
yxyx
== 33
== 217
217
yxyx
== 33
==yxyx ==33 kk
y = kxy = kx
The constant of variation is the steady rate of change. The constant k is the constant of variation, or the constant of proportionality.
The constant of variation is the steady rate of change. The constant k is the constant of variation, or the constant of proportionality.
Constant of VariationConstant of Variation
Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation.
Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation.
direct variation; k = 2.2direct variation; k = 2.2
f(x) = 2.2xf(x) = 2.2x
Example 2Example 2
This is not a direct variation; the variable must be a multiple of x.
This is not a direct variation; the variable must be a multiple of x.
y = 4x − 1y = 4x − 1
This is a direct variation; k = 45.This is a direct variation; k = 45.
d = 45td = 45t
This is not a direct variation; the coefficient of x must be positive.
This is not a direct variation; the coefficient of x must be positive.
y = −2xy = −2x
y = kx y = kx
y = mx + by = mx + b
Graph the direct variation y = 4x.Graph the direct variation y = 4x.
y
−4
0
4
x
−1
0
1
Example 3Example 3
xx
yy
Find k if y varies directly with x and y = 12 when x = . Write an equation for the direct variation.
Find k if y varies directly with x and y = 12 when x = . Write an equation for the direct variation.
1212
y = kxy = kx12 = k( )12 = k( )1
212
2(12) = k( )(2)2(12) = k( )(2)1212
k = 24k = 24y = 24xy = 24x
Example 4Example 4
If y varies directly with x and y = 6 when x = 2, find y when x = .
If y varies directly with x and y = 6 when x = 2, find y when x = .
y = kxy = kx
6 = k(2)6 = k(2)3 = k3 = k
y = 3xy = 3x
2323
y = 3( )y = 3( )2323
y = 2y = 2
Example 5Example 5
Find k if y varies directly with x and y = 14 when x = 4. Find k if y varies directly with x and y = 14 when x = 4.
k = 3.5k = 3.5
ExampleExample
Find k if y varies directly with x and y = 15 when x = 2. Find k if y varies directly with x and y = 15 when x = 2.
k = 7.5k = 7.5
ExampleExample
If y varies directly with x and y = 7 when x = 1, find y when x = 6.
If y varies directly with x and y = 7 when x = 1, find y when x = 6.
y = 42y = 42
ExampleExample
If y varies directly with x and y = 27 when x = 15, find y when x = 6.
If y varies directly with x and y = 27 when x = 15, find y when x = 6.
y =y = 545
545
ExampleExample
Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation. If not, explain.
Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation. If not, explain.
ExampleExample
Yes. k = 4.Yes. k = 4.y = 4xy = 4x
No. The y-intercept is not zero.No. The y-intercept is not zero.
y = 3x + 5y = 3x + 5
No. The slope is not positive.No. The slope is not positive.
y = −4xy = −4x
