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Understanding Direct Variation: Exploring the Relationship Between Variables

In algebra, direct variation describes a relationship between two variables, x and y, defined by the formula y = kx, where k is the constant of variation. For example, the price of hot dogs varies directly with the number purchased. If 7 hot dogs cost $21, we can derive the price per hot dog (k) by solving the equation. The resulting model allows us to predict the price for any quantity purchased. Direct variation graphs always pass through the origin, illustrating that no quantity means no cost. Explore examples and applications of this concept.

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Understanding Direct Variation: Exploring the Relationship Between Variables

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  1. Algebra 4.5 Direct Variation

  2. Direct Variation Model The two variables x and y are said to vary directly if their relationship is: y = kx k is the same as m (slope) k is called the constant of variation

  3. The price of hot dogs varies directly with the number of hotdogs you buy You buy hotdogs. x represents the number of hotdogs you buy. y represents the price you pay. y = kx Let’s figure out k, the price per hotdog. Suppose that when you buy 7 hotdogs, it costs $21. Plug that information into the model to solve for k. y = kx 21 = k(7) Now divide both sides by 7 to solve for k. 7 7 k = 3 The price per hotdog is $3. y = 3x You could use this model to find the price (y) for any number of hotdogs (x) you buy.

  4. y The graph of y = 3x goes through the origin. x All direct variation graphs go through the origin, because when x = 0, y= 0 also.

  5. y (price) y = 3x . (3,9) When you buy 3 hotdogs, you pay $9 . (2,6) When you buy 2 hotdogs, you pay $6 . (1,3) When you buy 1 hotdog, you pay $3 . x (number of hotdogs) (0,0) When you buy 0 hotdogs, you pay $0

  6. Finding the Constant of Variation (k) STEPS • Plug in the known values for x and y into the model: y = kx • Solve for k • Now write the model y = kx and replace k with the number • Use the model to find y for other values of x if needed

  7. Example • The variables x and y vary directly. When x = 24, y = 84. • Write the direct variation model that relates x and y. • Find y when x is 10.

  8. Example • The variables x and y vary directly. When x = 24, y = 84. • Write the direct variation model that relates x and y. • Find y when x is 10. 1.

  9. Example • The variables x and y vary directly. When x = 24, y = 84. • Write the direct variation model that relates x and y. • Find y when x is 10. 1.

  10. Example • The variables x and y vary directly. When x = 24, y = 84. • Write the direct variation model that relates x and y. • Find y when x is 10. 1.

  11. Example • The variables x and y vary directly. When x = 24, y = 84. • Write the direct variation model that relates x and y. • Find y when x is 10. 1. 2.

  12. Example • The variables x and y vary directly. When x = 24, y = 84. • Write the direct variation model that relates x and y. • Find y when x is 10. 1. 2. When x = 10, y = 35

  13. Example • The variables x and y vary directly. When x = ½, y = 18. • Write the direct variation model that relates x and y. • Find y when x is 5.

  14. Example • The variables x and y vary directly. When x = ½, y = 18. • Write the direct variation model that relates x and y. • Find y when x is 5. 1.

  15. Example • The variables x and y vary directly. When x = ½, y = 18. • Write the direct variation model that relates x and y. • Find y when x is 5. 1.

  16. Example • The variables x and y vary directly. When x = ½, y = 18. • Write the direct variation model that relates x and y. • Find y when x is 5. 1.

  17. Example • The variables x and y vary directly. When x = ½, y = 18. • Write the direct variation model that relates x and y. • Find y when x is 5. 1.

  18. Example • The variables x and y vary directly. When x = ½, y = 18. • Write the direct variation model that relates x and y. • Find y when x is 5. 1. 2.

  19. Example • The variables x and y vary directly. When x = ½, y = 18. • Write the direct variation model that relates x and y. • Find y when x is 5. 1. 2. When x = 5, y = 180

  20. Homework Pg. 238 #23-31, 38-41

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