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1.4 Before Now Why? Key Vocabulary • formula • solve for a variable Rewrite Formulas and Equations You solved equations. You will rewrite and evaluate formulas and equations. So you can apply geometric formulas, as in Ex. 36. A formula is an equation that relates two or more quantities, usually represented by variables. Some common formulas are shown below. Quantity READING The variables b1 and b2 are read as “b sub one” and “b sub two.” The small lowered numbers are called subscripts. Formula Distance d 5 rt Temperature F 5 } C 1 32 Area of a triangle Meaning of variables d 5 distance, r 5 rate, t 5 time 9 5 F 5 degrees Fahrenheit, C 5 degrees Celsius A 5 } bh 1 2 A 5 area, b 5 base, h 5 height Area of a rectangle A 5 lw A 5 area, l 5 length, w 5 width Perimeter of a rectangle P 5 2l 1 2w Area of a trapezoid A 5 } (b1 1 b2)h A 5 area, b1 5 one base, b2 5 other base, h 5 height Area of a circle A 5 πr2 A 5 area, r 5 radius Circumference of a circle C 5 2π r C 5 circumference, r 5 radius 1 2 P 5 perimeter, l 5 length, w 5 width To solve for a variable means to rewrite an equation as an equivalent equation in which the variable is on one side and does not appear on the other side. EXAMPLE 1 Rewrite a formula with two variables Solve the formula C 5 2p r for r. Then find the radius of a circle with a circumference of 44 inches. Solution STEP 1 Solve the formula for r. C 5 2πr Write circumference formula. C 2π }5r Divide each side by 2p. STEP 2 Substitute the given value into the rewritten formula. C 44 ø 7 r5} 5} 2π 2π Substitute 44 for C and simplify. c The radius of the circle is about 7 inches. 26 n2pe-0104.indd 26 Chapter 1 Equations and Inequalities 10/19/05 2:58:43 PM ✓ GUIDED PRACTICE for Example 1 1. Find the radius of a circle with a circumference of 25 feet. 2. The formula for the distance d between opposite vertices 2a where a is the distance of a regular hexagon is d 5 } } Ï3 d a between opposite sides. Solve the formula for a. Then find a when d 5 10 centimeters. EXAMPLE 2 Rewrite a formula with three variables Solve the formula P 5 2l 1 2w for w. Then find the width of a rectangle with a length of 12 meters and a perimeter of 41 meters. P 5 41 m 12 m Solution STEP 1 w Solve the formula for w. P 5 2l 1 2w Write perimeter formula. P 2 2l 5 2w Subtract 2l from each side. P 2 2l 2 Divide each side by 2. }5w STEP 2 Substitute the given values into the rewritten formula. 41 2 2(12) 2 w5} Substitute 41 for P and 12 for l. w 5 8.5 Simplify. c The width of the rectangle is 8.5 meters. "MHFCSB ✓ GUIDED PRACTICE at classzone.com for Example 2 3. Solve the formula P 5 2l 1 2w for l. Then find the length of a rectangle with a width of 7 inches and a perimeter of 30 inches. 4. Solve the formula A 5 lw for w. Then find the width of a rectangle with a length of 16 meters and an area of 40 square meters. Solve the formula for the variable in red. Then use the given information to find the value of the variable. 1 bh 5. A 5 } 2 1 bh 6. A 5 } 2 1 (b 1 b )h 7. A 5 } 2 2 1 b1 h h h b2 b b Find h if b 5 12 m and A 5 84 m 2. Find b if h 5 3 cm and A 5 9 cm 2. Find h if b1 5 6 in., b2 5 8 in., and A 5 70 in.2 1.4 Rewrite Formulas and Equations n2pe-0104.indd 27 27 10/19/05 2:58:46 PM REWRITING EQUATIONS The approach you use to solve a formula for a variable can be applied to other algebraic equations. EXAMPLE 3 Rewrite a linear equation Solve 9x 2 4y 5 7 for y. Then find the value of y when x 5 25. Solution Solve the equation for y. STEP 1 9x 2 4y 5 7 Write original equation. 24y 5 7 2 9x AVOID ERRORS Subtract 9x from each side. 7 1 9x y 5 2} } 4 When dividing each side of an equation by the same number, remember to divide every term by the number. Divide each side by 24. 4 STEP 2 Substitute the given value into the rewritten equation. 7 1 9 (25) y 5 2} } Substitute 25 for x. 7 2 45 y 5 2} } Multiply. y 5 213 Simplify. 4 4 4 4 9x 2 4y 5 7 CHECK Write original equation. 9(25) 2 4(213) 0 7 Substitute 25 for x and 213 for y. 757✓ EXAMPLE 4 Solution checks. Rewrite a nonlinear equation Solve 2y 1 xy 5 6 for y. Then find the value of y when x 5 23. Solution AVOID ERRORS STEP 1 If you rewrite the equation as 6 2 2y y5} x , then you have not solved for y because y still appears on both sides of the equation. Solve the equation for y. 2y 1 xy 5 6 Write original equation. (2 1 x)y 5 6 Distributive property 6 y5} 21x STEP 2 Substitute the given value into the rewritten equation. 6 y5} Substitute 23 for x. y 5 26 Simplify. 2 1 (23) ✓ Divide each side by (2 1 x). GUIDED PRACTICE for Examples 3 and 4 Solve the equation for y. Then find the value of y when x 5 2. 8. y 2 6x 5 7 11. 2x 1 5y 5 21 28 n2pe-0104.indd 28 9. 5y 2 x 5 13 10. 3x 1 2y 5 12 12. 3 5 2xy 2 x 13. 4y 2 xy 5 28 Chapter 1 Equations and Inequalities 10/19/05 2:58:47 PM EXAMPLE 5 Solve a multi-step problem MOVIE RENTAL A video store rents new movies for one price and older movies for a lower price, as shown at the right. • Write an equation that represents the store’s monthly revenue. • Solve the revenue equation for the variable representing the number of new movies rented. • The owner wants $12,000 in revenue per month. How many new movies must be rented if the number of older movies rented is 500? 1000? Solution STEP 1 Write a verbal model. Then write an equation. Monthly revenue (dollars) R Price of new movies 5 p Number of new movies (dollars/movie) 5 5 1 (movies) p n1 Price of older movies p (dollars/movie) 1 3 Number of older movies (movies) p n2 An equation is R 5 5n1 1 3n2. STEP 2 Solve the equation for n1. R 5 5n1 1 3n2 R 2 3n2 5 5n1 Write equation. Subtract 3n2 from each side. R 2 3n2 } 5 n1 5 Divide each side by 5. STEP 3 Calculate n1 for the given values of R and n2. 12,000 2 3 p 500 5 If n2 5 500, then n1 5 } 5 2100. 12,000 2 3 p 1000 5 If n2 5 1000, then n1 5 } 5 1800. c If 500 older movies are rented, then 2100 new movies must be rented. If 1000 older movies are rented, then 1800 new movies must be rented. ✓ GUIDED PRACTICE for Example 5 14. WHAT IF? In Example 5, how many new movies must be rented if the number of older movies rented is 1500? 15. WHAT IF? In Example 5, how many new movies must be rented if customers rent no older movies at all? 16. Solve the equation in Step 1 of Example 5 for n2. 1.4 Rewrite Formulas and Equations n2pe-0104.indd 29 29 10/19/05 2:58:48 PM 1.4 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 3, 9, and 35 ★ 5 STANDARDIZED TEST PRACTICE Exs. 2, 6, 15, 27, 36, and 38 SKILL PRACTICE 1. VOCABULARY Copy and complete: A(n) ? is an equation that relates two or more quantities. 2. ★ WRITING What does it mean to solve for a variable in an equation? EXAMPLES 1 and 2 on pp. 26–27 for Exs. 3–6 REWRITING FORMULAS Solve the formula for the indicated variable. Then use the given information to find the value of the variable. 3. Solve A 5 lw for l. Then find the length of a rectangle with a width of 50 millimeters and an area of 250 square millimeters. 1 bh for b. Then find the base of a triangle with a height of 4. Solve A 5 } 2 6 inches and an area of 24 square inches. 1 (b 1 b )h for h. Then find the height of a trapezoid with 5. Solve A 5 } 2 2 1 bases of lengths 10 centimeters and 15 centimeters and an area of 75 square centimeters. 6. ★ MULTIPLE CHOICE What equation do you obtain when you solve the 1 (b 1 b )h for b ? formula A 5 } 2 1 2 1 2A 2 b A b1 5 } 2 A 2b B b1 5 } 2 C b1 5 2A 2 b2h 2A D b1 5 } h 2h h 2 b2 EXAMPLE 3 REWRITING EQUATIONS Solve the equation for y. Then find the value of y for the on p. 28 for Exs. 7–17 given value of x. 7. 3x 1 y 5 26; x 5 7 8. 4y 1 x 5 24; x 5 8 9. 6x 1 5y 5 31; x 5 24 10. 15x 1 4y 5 9; x 5 23 11. 9x 2 6y 5 63; x 5 5 12. 10x 2 18y 5 84; x 5 6 13. 8y 2 14x 5 222; x 5 5 14. 9y 2 4x 5 230; x 5 8 15. ★ MULTIPLE CHOICE What equation do you obtain when you solve the equation 4x 2 5y 5 20 for y? 5y 1 5 A x5} 4 4x 1 4 B y 5 2} 5 4x 2 4 C y5} 4 x 2 20 D y5} 5 5 ERROR ANALYSIS Describe and correct the error in solving the equation for y. 16. 17. 27x 1 5y 5 2 5y 5 7x 1 2 4y 5 9 1 xy 7x 1 2 y5} y5} 5 30 n2pe-0104.indd 30 4y 2 xy 5 9 9 1 xy 4 Chapter 1 Equations and Inequalities 10/19/05 2:58:50 PM GEOMETRY Solve the formula for the variable in red. Then use the given information to find the value of the variable. Round to the nearest tenth. 18. Area of a circular ring 19. Lateral surface area of a truncated cylinder S 5 πr(h 1 k) A 5 2πrw 20. Volume of an ellipsoid 4 πabc V5} 3 b w r a c k h r Find r if w 5 4 ft and A 5 120 ft 2. Find h if r 5 2 cm, k 5 3 cm, and S 5 50 cm 2. Find c if a 5 4 in., b 5 3 in., and V 5 60 in.3 EXAMPLE 4 REWRITING EQUATIONS Solve the equation for y. Then find the value of y for the on p. 28 for Exs. 21–26 given value of x. 21. xy 2 3x 5 40; x 5 5 22. 7x 2 xy 5 218; x 5 24 23. 3xy 2 28 5 16x; x 5 4 24. 9y 1 6xy 5 30; x 5 26 25. y 2 2xy 5 15; x 5 21 26. 4x 1 7y 1 5xy 5 0; x 5 1 27. ★ SHORT RESPONSE Consider the equation 15x 2 9y 5 27. To find the value of y when x 5 2, you can use two methods. Method 1 Solve the original equation for y and then substitute 2 for x. Method 2 Substitute 2 for x and then solve the resulting equation for y. Show the steps of the two methods. Which method is more efficient if you need to find the value of y for several values of x? Explain. REASONING Solve for the indicated variable. 28. Solve xy 5 x 1 y for y. 29. Solve xyz 5 x 1 y 1 z for z. 1 1 1 5 1 for y. 30. Solve } } x y 1 1 1 1 1 5 1 for z. 31. Solve } } } x y z 32. CHALLENGE Write a formula giving the area of a circle in terms of its circumference. PROBLEM SOLVING EXAMPLE 5 on p. 29 for Exs. 33–38 33. TREE DIAMETER You can estimate the diameter of a tree without boring through it by measuring its circumference. Solve the formula C 5 πd for d. Then find the diameter of an oak that has a circumference of 113 inches. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN 34. DESIGN The fabric panels on a kite are rhombuses. A formula for the length } of the long diagonal d is d 5 sÏ 3 where s is the length of a side. Solve the formula for s. Then find the value of s when d 5 15 inches. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN s s d s 1.4 Rewrite Formulas and Equations n2pe-0104.indd 31 s 31 10/19/05 2:58:51 PM 35. TEMPERATURE The formula for converting temperatures from degrees 9 C 1 32. Solve the formula for C. Celsius to degrees Fahrenheit is F 5 } 5 Then find the temperature in degrees Celsius that corresponds to 508F. 36. ★ EXTENDED RESPONSE A quarter mile running track is shaped as shown. The formula for the inside perimeter is P 5 2πr 1 2x. a. Solve the perimeter formula for r. r r b. For a quarter mile track, P 5 440 yards. Find r when x 5 75 yards, 100 yards, 120 yards, and 150 yards. c. What are the greatest and least possible values of r if x P 5 440 yards? Explain how you found the values, and sketch the track corresponding to each extreme value. 37. MULTI-STEP PROBLEM A tuxedo shop rents classic tuxedos for $80 and designer tuxedos for $150. Write an equation that represents the shop’s revenue. Solve the equation for the variable representing the number of designer tuxedos rented. The shop owner wants $60,000 in revenue during prom season. How many designer tuxedos must be rented if the number of classic tuxedos rented is 600? 450? 300? 38. ★ OPEN-ENDED MATH The volume of a donut-like shape called a torus is given by the formula V 5 2π 2r 2 R where r and R are the radii shown and r ≤ R. r R r R a. Solve the formula for R. b. A metal ring in the shape of a torus has a volume of w 100 cubic centimeters. Choose three possible values of r, and find the corresponding values of R. l 39. CHALLENGE A rectangular piece of paper with length l and width w can be rolled to form the lateral surface of a cylinder in two ways, assuming no overlapping. Write a formula for the volume of each cylinder in terms of l and w. l w MIXED REVIEW PREVIEW Write an expression to answer the question. (p. 984) Prepare for Lesson 1.5 in Exs. 40–47. 40. You have $250 in a bank account and deposit x dollars. What is your current balance? 41. You buy x CDs for $12.99 each. How much do you spend? Evaluate the expression for the given value of the variable. (p. 10) 42. 6j 1 8 when j 5 23 43. 6 1 4k 4 2 when k 5 3 44. 8g 2 8g p 2 when g 5 21 45. 25m3 1 m2 when m 5 10 46. (n 1 7)2 2 4 when n 5 2 47. (3p 2 17) 3 when p 5 5 Solve the equation. Check your solution. (p. 18) 32 n2pe-0104.indd 32 48. 4x 1 7 5 10x 1 25 49. 15 2 2y 5 2y 2 45 50. 56 5 4(4 1 2z) 51. 9(p 1 3) 5 35p 1 1 5q 2 9 5 1 52. } 3 1r 1 1r 5 5 53. } } 4 6 Chapter 1 EXTRA EquationsPRACTICE and Inequalities for Lesson 1.4, p. 1010 ONLINE QUIZ at classzone.com 10/19/05 2:58:54 PM MIXED REVIEW of Problem Solving STATE TEST PRACTICE classzone.com Lessons 1.1–1.4 1. MULTI-STEP PROBLEM There is a $50 annual membership fee to join an urban car rental service. Using a car costs $8.50 per hour. a. Write a verbal model for this situation. Then use the verbal model to write an algebraic expression. b. How much will it cost to join the service and drive for 20 hours? 2. MULTI-STEP PROBLEM You are attending a museum. You have $50 to spend. Admission to the museum is $15. Admission to each special exhibit inside the museum is $10. 5. GRIDDED ANSWER You drive from Chicago to St. Louis, a distance of 290 miles. Your average speed is 60 miles per hour. How many hours does the trip take? Round your answer to the nearest tenth of an hour. 6. OPEN-ENDED Describe a shopping situation that can be modeled by the equation 10x 1 29y 5 78. 7. EXTENDED RESPONSE In one year, the Bureau of Engraving and Printing printed $10 and $20 bills with a total value of $66,368,000. The total number of $10 and $20 bills was 3,577,600. a. Write an equation that can be used to find the number of special exhibits you can include in your visit. b. Solve the equation. Interpret your answer in terms of the problem. 3. SHORT RESPONSE In hockey, each player has a statistic called plus/minus, which is the difference between the number of goals scored by the player’s team and the number of goals scored by the other team when the player is on the ice. List the players shown in order from least to greatest plus/minus. Whose plus/ minus score is best? Explain. Player Number Value $10 bills x 10x $20 bills ? ? Total 3,577,600 66,368,000 a. Copy and complete the table. b. Write and solve an equation to find how many $10 bills and how many $20 bills were printed. c. Compare the total value of the $10 bills printed with the total value of the $20 bills printed. Plus/Minus Vincent Lecavalier 23 Dave Andreychuk 29 Ruslan Fedotenko 14 Martin St. Louis 35 Cory Sarich 5 Tim Taylor 25 4. SHORT RESPONSE You are in charge of buying food for a school picnic. You have $45 to spend on ground beef and chicken. Ground beef costs $1.80 per pound and chicken costs $1.00 per pound. Write an equation representing the situation. You want to buy equal amounts of ground beef and chicken. How much of each can you buy? Show how you found your answer. 8. OPEN-ENDED You have two summer jobs. You mow lawns for $20 per lawn. You also work at a restaurant for $7.50 per hour. Write an equation for the total amount of money you earn. Then find three different ways to earn $300 during one week. 9. GRIDDED ANSWER The liopleurodon, a swimming dinosaur from the Late Jurassic period, could grow to 25 meters in length. Use the fact that 1 in. 5 2.54 cm to convert the length to feet. Round your answer to the nearest foot. 10. GRIDDED ANSWER The formula for the volume 1 Bh. Find h (in centimeters) if of a cone is V 5 } 3 V 5 176 cm3 and B 5 40 cm 2. Mixed Review of Problem Solving n2pe-0104.indd 33 33 10/19/05 2:58:57 PM