11-1 Study Guide and Intervention Inverse Variation Identify and Use Inverse Variations An inverse variation is an equation in the form of = or = If two points y xy k. (, ) and (, ) are solutions of an inverse variation, then = and = k k. ⋅ ⋅ Product Rule for Inverse Variation = ⋅ ⋅ From the product rule, you can form the proportion = . Example: If varies inversely as and = 12 when = 4, find when = 18. y x y x x y Method 1 Use the product rule. Use a proportion. Method 2 = Product rule for inverse variation ⋅ ⋅ = Proportion for inverse variation 4 12 = 18 = 4, = 12, = 18 ⋅ ⋅ = = 4, = 12, = 18 = Divide each side by 18. 48 = 18 Cross multiply. = Simplify. = Simplify. Both methods show that = when = 18. y Exercises Determine whether each table or equation represents an or a variation. Explain. inverse direct 1. 2. = 6x 3. = 15 y xy x y 3 6 5 10 8 16 12 24 Assume that varies inversely as Write an inverse variation equation that relates and Then solve. y x. x y. 4. If = 10 when = 5, 5. If = 8 when = y x y x –2, find when = 2. find when = 4. y x y x 6. If = 100 when = 120, 7. If = when = 4, y x y x –16 find when = 20. find when = 32. x y x y 8. If = when = 25, find when = 5. y x y x –7.5 9. DRIVING The Gerardi family can travel to Oshkosh, Wisconsin, from Chicago, Illinois, in 4 if they drive an average of 45 miles per hour. How long would it take them if they increased their average speed to 50 miles per hour? 10. GEOMETRY For a rectangle with given area, the width of the rectangle varies inversely as the length. If the width of the rectangle is 40 meters when the length is 5 meters, find the width of the rectangle when the length is 20 meters.
11-1 Study Guide and Intervention Inverse Variation Identify and Use Inverse Variations An inverse variation is an equation in the form of y = or xy = k. If two points (, ) and (, ) are solutions of an inverse variation, then ⋅ = k and ⋅ = k. Product Rule for Inverse Variation ⋅ = ⋅ From the product rule, you can form the proportion = . Example: If y varies inversely as x and y = 12 when x = 4, find x when y = 18. Method 1 Use the product rule. ⋅ = ⋅ Product rule for inverse variation 4 ⋅ 12 = ⋅ 18 = 4, = 12, = 18 = Divide each side by 18. = Simplify. Method 2 Use a proportion. = Proportion for inverse variation = = 4, = 12, = 18 48 = 18 Cross multiply. = Simplify. Both methods show that = when y = 18. Exercises Determine whether each table or equation represents an inverse or a direct variation. Explain. x y 3 6 5 10 8 16 12 24 1. 2. y = 6x 3. xy = 15 Assume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then solve. 4. If y = 10 when x = 5, 5. If y = 8 when x = –2, find y when x = 2. find y when x = 4. 6. If y = 100 when x = 120, 7. If y = –16 when x = 4, find x when y = 20. find x when y = 32. 8. If y = –7.5 when x = 25, find y when x = 5. 9. DRIVING The Gerardi family can travel to Oshkosh, Wisconsin, from Chicago, Illinois, in 4 if they drive an average of 45 miles per hour. How long would it take them if they increased their average speed to 50 miles per hour? 10. GEOMETRY For a rectangle with given area, the width of the rectangle varies inversely as the length. If the width of the rectangle is 40 meters when the length is 5 meters, find the width of the rectangle when the length is 20 meters.
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