ALGEBRA 1 LESSON 8 -10 Inverse Variation (For help, go to Lesson 5 -5. ) Suppose y varies directly with x. Find each constant of variation. 1. y = 5 x 2. y = – 7 x 3. 3 y = x 4. 0. 25 y = x Write an equation of the direct variation that includes the given point. 5. (2, 4) 6. (3, 1. 5) 7. (– 4, 1) 8 -10 8. (– 5, – 2)
ALGEBRA 1 LESSON 8 -10 Inverse Variation Solutions 1. y = 5 x; constant of variation = 5 2. y = – 7 x; constant of variation = – 7 3. 3 y = x 1 1 • 3 y = • x 3 3 1 3 y = x; constant of variation = 1 3 4. 0. 25 y = x 1 y = x 4 1 4 4 • y = 4 • x y = 4 x; constant of variation = 4 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation Solutions (continued) 5. Point (2, 4) in y = kx: 4 = k(2), so k = 2 and y = 2 x. 6. Point (3, 1. 5) in y = kx: 1. 5 = k(3), so k = 0. 5 and y = 0. 5 x. 1 1 7. Point (– 4, 1) in y = kx: 1 = k(– 4), so k = – and y = – x. 4 4 2 5 8. Point (– 5, – 2) in y = kx: – 2 = k(– 5), so k = and y = x. 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation Suppose y varies inversely with x, and y = 9 when x = 8. Write an equation for the inverse variation. xy = k (8)(9) = k Use the general form for an inverse variation. Substitute 8 for x and 9 for y. 72 = k Multiply to solve for k. xy = 72 Write an equation. Substitute 72 for k in xy = k. 72 The equation of the inverse variation is xy = 72 or y = . x 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation The points (5, 6) and (3, y) are two points on the graph of an inverse variation. Find the missing value. x 1 • y 1 = x 2 • y 2 5(6) = 3 y 2 Use the equation x 1 • y 1 = x 2 • y 2 since you know coordinates, but not the constant of variation. Substitute 5 for x 1, 6 for y 1, and 3 for x 2. 30 = 3 y 2 Simplify. 10 = y 2 Solve for y 2. The missing value is 10. The point (3, 10) is on the graph of the inverse variation that includes the point (5, 6). 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation Jeff weighs 130 pounds and is 5 ft from the lever’s fulcrum. If Tracy weighs 93 pounds, how far from the fulcrum should she sit in order to balance the lever? Relate: A weight of 130 lb is 5 ft from the fulcrum. A weight of 93 lb is x ft from the fulcrum. Weight and distance vary inversely. Define: Let weight 1 = 130 lb Let weight 2 = 93 lb Let distance 1 = 5 ft Let distance 2 = x ft 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation (continued) Write: weight 1 • distance 1 = weight 2 • distance 2 130 • 5 = 93 • x Substitute. 650 = 93 x Simplify. 650 93 = x Solve for x. 6. 99 = x Simplify. Tracy should sit 6. 99, or 7 ft, from the fulcrum to balance the lever. 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation Decide if each data set represents a direct variation or an inverse variation. Then write an equation to model the data. a. x 3 5 10 y 10 6 3 The values of y seem to vary inversely with the values of x. Check each product xy. xy: 3(10) = 30 5(6) = 30 10(3) = 30 The product of xy is the same for all pairs of data. So, this is an inverse variation, and k = 30. The equation is xy = 30. 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation (continued) b. y x x 2 4 8 y 3 6 12 3 = 1. 5 2 The values of y seem to vary directly with the values of x. y Check each ratio . x 6 = 1. 5 4 y The ratio is the same for all pairs of data. x So, this is a direct variation, and k = 1. 5. The equation is y = 1. 5 x. 8 -10 12 = 1. 5 8
ALGEBRA 1 LESSON 8 -10 Inverse Variation Explain whether each situation represents a direct variation or an inverse variation. a. You buy several souvenirs for $10 each. The cost per souvenir times the number of souvenirs equals the total cost of the souvenirs. cost Since the ratio is constant at $10 each, this is a direct variation. souvenirs b. The cost of a $25 birthday present is split among several friends. The cost person times the number of people equals the total cost of the gift. Since the total cost is a constant product of $25, this is an inverse variation. 8 -10
ALGEBRA 1 LESSON 8 -10 Inverse Variation 1. The points (5, 1) and (10, y) are on the graph of an inverse variation. Find y. 0. 5 2. Find the constant of variation k for the inverse variation where a = 2. 5 when b = 7. 17. 5 y 3. Write an equation to model the data x 1 and complete the table. 1 3 2 1 xy = 3 3 6 1 9 1 18 4. Tell whether each situation represents a direct variation or an inverse variation. a. You buy several notebooks for $3 each. direct variation b. The $45 cost of a dinner at a restaurant is split among several people. Inverse variation 8 -10
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