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**Lesson 5-5 Direct Variation**

Algebra I, Ms. Turk

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**Definition: direct variation**

A direct variation is a function in the form y = kx where k does not equal 0.

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**Definition: constant of variation**

The constant of variation is k, the coefficient of x.

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**An equation is a direct variation if:**

its graph is a line that passes through zero, or the equation can be written in the form y = kx.

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**Is an Equation a Direct Variation**

Is an Equation a Direct Variation? If it is, find the constant of variation. 5x + 2y = 0 Solve for y. 1. Subtract 5x. 2. Divide by 2. Yes, it’s a direct variation. Constant of variable, k, is

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**Is an Equation a Direct Variation**

Is an Equation a Direct Variation? If it is, find the constant of variation. 5x + 2y = 9 Solve for y. 1. Subtract 5x. 2. Divide by 2. No, it’s not a direct variation. It’s not in the form y = kx.

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**Is an Equation a Direct Variation**

Is an Equation a Direct Variation? If it is, find the constant of variation. 7y = 2x Yes, it is a direct variation. The constant of variation, k, is

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Writing an Equation Given a Point Write an equation of the direct variation that includes the point (4, -3). y = kx Start with the function form of the direct variation. -3 = k(4) Substitute 4 for x and -3 for y. Divide by 4 to solve for k. Substitute the value of k into the original formula. The Answer!

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Writing an Equation Given a Point Write an equation of the direct variation that includes the point (-3, -6). y = kx Start with the function form of the direct variation. -6 = k(-3) Substitute -3 for x and -6 for y. Divide by -3 to solve for k. Substitute the value of k into the original formula. The Answer!

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**Real-World Problem Solving**

Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see the lightning, you are about 2 miles from the lightning. Write an equation for the relationship between time and distance.

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**Real-World Problem Solving**

Relate: The distance varies directly with the time. When x = 10, y = 2. Define: Let x = number of seconds between seeing lightning and hearing thunder. Let y = distance in miles from lightning.

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**Real-World Problem Solving**

y = kx Use general form of direct variation. 2 = k(10) Substitute 2 for y and 10 for x. Solve for k. Write an equation using the value for k.

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**Real-World Problem Solving**

A recipe for a dozen corn muffins calls for 1 cup of flower. The number of muffins varies directly with the amount of flour you use. Write a direct variation for the relationship between the number of cups of flour and the number of muffins.

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**Real-World Problem Solving**

y = kx 12 = k(1) k = 12 y = 12x Let y = 12 (1 dozen) Let x = 1

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**Real-World Problem Solving**

The force you must apply to lift an object varies directly with the object’s weight. You would need to apply lb of force to a windlass to lift a 28-lb weight. How much force would you need to lift 100 lb? Relate: A force of lifts 28 lb. What lifts 100 lb?

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**You need about 2.2 lb of force to lift 100 lb.**

Use a proportion. Cross multiply n = 62.5 Solve for n. n ≈ 2.2 You need about 2.2 lb of force to lift 100 lb.

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**Real-World Problem Solving**

Suppose a second windlass requires 0.5 lb of force to lift an object that weighs 32 lb. How much force would you need to lift 160 lb?

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**You need 2.5 lb of force to lift 160 lb.**

Use a proportion. Cross multiply n = 80 Solve for n. n = 2.5 You need 2.5 lb of force to lift 160 lb.

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