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5-5: Direct Variation

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**What’s Direct Variation?**

Direct variation is a function where y = kx, where k ≠ 0 The variables y and x are vary directly with each other, where k is the constant of variation

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**What’s Direct Variation?**

To put simply: In a direct variation, when one value increases, the other also increases. (So in the equation y = kx, when y increases, x also increases)

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**Identify The equation is a direct variation when…**

- it can be written in the form of y = kx

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Example 1 Is the equation a direct variation? If it is, find the constant of variation. y 7.5x = 0

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y – 7.5x = 0 y – 7.5x + 7.5x = x y = 7.5x YES, it is a direct variation because it can be written as the form y = kx, the constant of variation (k) = 7.5

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**Quick Check 6y = 12x 7y = 3x + 4 6y = 12x 7y = 3x + 4 12 y = 2x**

YES, it’s a direct variation, k = 2 NO, it’s NOT a direct variation 6y = 12x 7y = 3x + 4 6 6 7 7 y = x y = x + 6 7 7

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Example 2 Write an equation of the direct variation that includes the given point. (5,1) Start with the function form Substitute (5,1) with (x,y) Solve for k Substitute 1/5 for k y = kx 1 = k(5) k = 1/5 y = 1/5x

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**Quick Check (4, 16) (3, 2) (4, 16) (3, 2) y = kx 16 = k(4) 4 = k**

y = x 3

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Example 3 Tony works at a Pizza store, his pay (n) varies directly with his hours of work (w). On Saturday, Tony worked for 3 hours at the store, and his hourly pay is 20$. Answer the following questions. Write an equation of direct variation for Tony’s pay and his hours of work. What is Tony’s pay on Saturday? What will the graph of this problem look like?

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**Example 3 (Answers) a) y = kx n = 20$w b) n = 20$(3) n = 60$**

c) The graph will be positive, since in a direct variation, if one variable increases, the other also increases.

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Example 3 Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see lightning, you are about 2 miles from the lightning. Write an equation for the relationship between time and distance.

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**Relate: The distance varies directly with the time. When x = 10, y = 2**

Define: Let x = the # of seconds between seeing lightning and hearing thunder Let y = distance in miles from the lightning y = kx 2 = k(10) 10 10 = k y = x 5 5

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**Quick Check Let x = the number of hours Let y = the amount of money**

If you work for 5 hours, you’ll get $90. Write a direct variation for the relationship between the number of hours and the amount of money. y = kx 90 = k(5) 18 = k y = 18x 5 5

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**x y y/x x y Example 4 -5 -10 -10/-5 = 2 2 -4 -4/2 = 2 12 12/-4 = -3**

For each table, use the ratio y/x to tell whether y varies directly with x. If it does, write an equation for the direct variation x y y/x -5 -10 -10/-5 = 2 2 -4 -4/2 = 2 12 12/-4 = -3 x y -10 -5 -4 2 12 No, the ratio y/x is not the same for all pairs of data

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x y y/x 7 14 1 2 -4 -8 y/x 14/7 = 2 2/1 = 2 -8/-4 = 2 Yes, the constant of variation is 2. The equation is y = 2x

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**x y y/x y/x Quick Check 7 -21 22 -66 -5 15 -21/7 = -3 -66/22 = -3**

15/-5 = -3 Yes, the constant of variation is -3. The equation is y = -3x

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THE END

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**…Or Not Next we’ll play Jeopardy**

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