Exercise Solve. 5 = 3x x = 5 3

Exercise Solve. 4 = k 1 3 k = 12

Exercise Solve. y 25 3 5 = y = 15

Exercise Solve. 3 5 25 x = x = 41 2 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 y x 1 30 2 60 3 90 4 120

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 y 1 3 3 9 5 15 7 21

x y 1 3 3 9 5 15 7 21 = 3 1 y x 3 = 21 7 y x 3 = 9 3 y x 3 = y x 3 k y = kx = 15 5 y x 3 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 −1 1 y −4 4

y x

Example 4 Find k if y varies directly with x and y = 12 when x = . Write an equation for the direct variation. 1 2 y = kx k = 24 12 = k( ) 1 2 y = 24x 2(12) = k( )(2) 1 2

Example 5 If y varies directly with x and y = 6 when x = 2, find y when x = . 2 3 y = kx y = 3x y = 3( ) 2 3 6 = k(2) 3 = k y = 2

Example Find k if y varies directly with x and y = 14 when x = 4.

Example Find k if y varies directly with x and y = 15 when x = 2.

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 5 y =

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.