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Published byHailey Nickelson Modified over 2 years ago

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Solve. Exercise 5 = 3x x =

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4 = k Solve. k = 12 Exercise

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= = y 25 Solve. y = Exercise

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= = 25 x Solve x = Exercise

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If 4 = 6k, what is the value of 3k? 2 2 Exercise

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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 Variation

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Variables are directly proportional when y is said to vary directly with x. Directly Proportional

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The constant k is the constant of variation, or the constant of proportionality. Constant of Proportionality

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x hoursy miles yxyx

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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 Example 1

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x y y = 3x = = yxyx yxyx = = 3 3 = = yxyx yxyx = = 3 3 = = 15 5 yxyx yxyx = = 3 3 = = 21 7 yxyx yxyx = = 3 3 = = yxyx yxyx = = 3 3 k k y = kx

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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 Variation

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Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation. direct variation; k = 2.2 f(x) = 2.2x Example 2

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This is not a direct variation; the variable must be a multiple of x. y = 4x − 1 This is a direct variation; k = 45. d = 45t This is not a direct variation; the coefficient of x must be positive. y = −2x

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y = kx y = mx + b

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Graph the direct variation y = 4x. y −4 0 4 x −1 0 1 Example 3

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x x y y

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

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

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Find k if y varies directly with x and y = 14 when x = 4. k = 3.5 Example

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Find k if y varies directly with x and y = 15 when x = 2. k = 7.5 Example

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If y varies directly with x and y = 7 when x = 1, find y when x = 6. y = 42 Example

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If y varies directly with x and y = 27 when x = 15, find y when x = 6. y = 54 5 Example

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Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation. If not, explain. Example

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Yes. k = 4. y = 4x No. The y-intercept is not zero. y = 3x + 5 No. The slope is not positive. y = −4x

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