Page 1 of 7 1.4 Rewiting Equations and Fomulas What you should lean GOAL 1 Rewite equations with moe than one vaiale. GOAL Rewite common fomulas, as applied in Example 5. Why you should lean it To solve eal-life polems, such as finding how much you should chage fo tickets to a enefit concet in Example 4. GOAL 1 EQUATIONS WITH MORE THAN ONE VARIABLE In Lesson 1.3 you solved equations with one vaiale. Many equations involve moe than one vaiale. You can solve such an equation fo one of its vaiales. EXAMPLE 1 Solve 7x º3y = 8 fo y. 7x º3y = 8 Rewiting an Equation with Moe Than One Vaiale Wite oiginal equation. º3y = º7x + 8 Sutact 7x fom each side. y = 7 3 x º 8 Divide each side y º3. 3 ACTIVITY Developing Concepts Equations with Moe Than One Vaiale Given the equation x +5y = 4, use each method elow to find y when x = º3, º1,, and 6. Tell which method is moe efficient. Method 1 Sustitute x = º3 into x +5y = 4 and solve fo y. Repeat this pocess fo the othe values of x. Method Solve x + 5y = 4 fo y. Then evaluate the esulting expession fo y using each of the given values of x. EXAMPLE Calculating the Value of a Vaiale Given the equation x + xy = 1, find the value of y when x = º1 and x = 3. Solve the equation fo y. x + xy = 1 Wite oiginal equation. xy = 1 º x Sutact x fom each side. y = 1º x Divide each side y x. x Then calculate the value of y fo each value of x. When x = º1: y = 1º ( º º1) 1 = º When x = 3: y = 1º 3 3 = º 3 6 Chapte 1 Equations and Inequalities
Page of 7 EXAMPLE 3 Witing an Equation with Moe Than One Vaiale Benefit Concet You ae oganizing a enefit concet. You plan on having only two types of tickets: adult and child. Wite an equation with moe than one vaiale that epesents the evenue fom the concet. How many vaiales ae in you equation? PROBLEM SOLVING STRATEGY VERBAL MODEL Total evenue = Adult Nume + Child ticket pice of adults ticket pice Nume of childen LABELS Total evenue = R Adult ticket pice = p 1 Nume of adults = A Child ticket pice = p Nume of childen = C (dollas) (dollas pe adult) (adults) (dollas pe child) (childen) ALGEBRAIC MODEL R = A + C p 1 p This equation has five vaiales. The vaiales p 1 and p ae ead as p su one and p su two. The small loweed numes 1 and ae suscipts used to indicate the two diffeent pice vaiales. EXAMPLE 4 Using an Equation with Moe Than One Vaiale BENEFIT CONCERT Fo the concet in Example 3, you goal is to sell $5,000 in tickets. You plan to chage $5.5 pe adult and expect to sell 800 adult tickets. You need to detemine what to chage fo child tickets. How much should you chage pe child if you expect to sell 00 child tickets? 300 child tickets? 400 child tickets? FOCUS ON APPLICATIONS Fist solve the equation R = p 1 A + p C fom Example 3 fo p. R = p 1 A + p C Wite oiginal equation. R º p 1 A = p C R º p 1 A C Sutact p 1 A fom each side. = p Divide each side y C. Now sustitute the known values of the vaiales into the equation. BENEFIT CONCERT Fam Aid, a type of enefit concet, egan in 1985. Since that time Fam Aid has distiuted moe than $13,000,000 to family fams thoughout the United States. 5,000 º 5.5(800) If C = 00, the child ticket pice is p = = $4. 00 5,000 º 5.5(800) If C = 300, the child ticket pice is p = = $16. 300 5,000 º 5.5(800) If C = 400, the child ticket pice is p = = $1. 400 1.4 Rewiting Equations and Fomulas 7
Page 3 of 7 GOAL REWRITING COMMON FORMULAS Thoughout this couse you will e using many fomulas. Seveal ae listed elow. COMMON FORMULAS FORMULA VARIABLES Distance d = t d = distance, = ate, t = time Simple Inteest I = Pt I = inteest, P = pincipal, = ate, t = time Tempeatue F = 9 C + 3 F = degees Fahenheit, C = degees Celsius 5 Aea of Tiangle A = 1 h A = aea, = ase, h = height Aea of Rectangle A = w A = aea, = length, w = width Peimete of Rectangle P = + w P = peimete, = length, w = width Aea of Tapezoid A = 1 ( 1 + )h A = aea, 1 = one ase, = othe ase, h = height Aea of Cicle A = π A = aea, = adius Cicumfeence of Cicle C = π C = cicumfeence, = adius EXAMPLE 5 Rewiting a Common Fomula Skills Review Fo help with peimete, see p. 914. The fomula fo the peimete of a ectangle is P = + w. Solve fo w. P = + w Wite peimete fomula. P º = w Sutact fom each side. P º = w Divide each side y. Gadening EXAMPLE 6 Applying a Common Fomula You have 40 feet of fencing with which to enclose a ectangula gaden. Expess the gaden s aea in tems of its length only. Use the fomula fo the aea of a ectangle, A = w, and the esult of Example 5. A = w Wite aea fomula. Sustitute } P º A = P º } fo w. A = 40 º Sustitute 40 fo P. A = (0 º ) Simplify. 8 Chapte 1 Equations and Inequalities
Page 4 of 7 GUIDED PRACTICE Vocaulay Check Concept Check Skill Check 1. Complete this statement: A = w is an example of a(n)?.. Which of the following ae equations with moe than one vaiale? A. x + 5 = 9 º 5x B. 4x + 10y = 6 C. x º 8 = 3y + 7 3. Use the equation fom Example 3. Descie how you would solve fo A. Solve the equation fo y. 4. 4x + 8y = 17 5. 5x º 3y = 9 6. 5y º 3x = 15 7. 3 4 x + 5y = 0 8. xy + x = 8 9. 3 x º 1 y = 1 In Execises 10 and 11, use the following infomation. The aea A of an ellipse is given y the fomula A = πa whee a and ae half the lengths of the majo and mino axes. (The longe chod is the majo axis.) 10. Solve the fomula fo a. 11. Use the esult fom Execise 10 to find the length of the majo axis of an ellipse whose aea is 157 squae inches and whose mino axis is 10 inches long. (Use 3.14 fo π.) a a PRACTICE AND APPLICATIONS Exta Pactice to help you maste skills is on p. 940. HOMEWORK HELP Examples 1, : Exs. 1 3 Examples 3, 4: Exs. 33 39 Examples 5, 6: Exs. 4 3, 40 4 EXPLORING METHODS Find the value of y fo the given value of x using two methods. Fist, sustitute the value of x into the equation and then solve fo y. Second, solve fo y and then sustitute the value of x into the equation. 1. 4x + 9y = 30; x = 3 13. 5x º 7y = 1; x = 1 14. xy + 3x = 5; x = 5 15. 9y º 4x = º16; x = 8 16. ºy º x = º11; x = º4 17. ºx = 3y º 55; x = 0 18. x = 4 + xy; x = º1 19. ºxy + 3x = 30; x = 15 0. º4x + 7y + 7 = 0; x = 7 1. 6x º 5y º 44 = 0; x = 4. 1 x º 4 5 y = 19; x = 6 3. 3 4 x = º 9 y + 1; x = 10 11 REWRITING FORMULAS Solve the fomula fo the indicated vaiale. 4. Cicumfeence of a Cicle 5. Volume of a Cone Solve fo : C = π Solve fo h: V = 1 3 π h 6. Aea of a Tiangle 7. Investment at Simple Inteest Solve fo : A = 1 h Solve fo P: I = Pt 8. Celsius to Fahenheit 9. Aea of a Tapezoid Solve fo C: F = 9 5 C + 3 Solve fo : A = 1 ( 1 + )h 1.4 Rewiting Equations and Fomulas 9
Page 5 of 7 GEOMETRY CONNECTION In Execises 30º3, solve the fomula fo the indicated vaiale. Then evaluate the ewitten fomula fo the given values. (Include units of measue in you answe.) 30. Aea of a cicula 31. Suface aea of a 3. Peimete of a tack: ing: A = πpw cylinde: P = π + x Solve fo p. Find p S = πh + π Solve fo. Find when when A = cm Solve fo h. Find h P = 440 yd and and w = cm. when S = 105 in. x = 110 yd. and = 3 in. h x HONEYBEES In Execises 33 and 34, use the following infomation. A foage honeyee spends aout thee weeks ecoming accustomed to the immediate suoundings of its hive and spends the est of its life collecting pollen and necta. The total nume of miles T a foage honeyee flies in its lifetime L (in days) can e modeled y T = m(l º 1) whee m is the nume of miles it flies each day. 33. Solve the equation T = m(l º 1) fo L. 34. A foage honeyee s flight muscles last only aout 500 miles; afte that the ee dies. Some foage honeyees fly aout 55 miles pe day. Appoximately how many days do these ees live? SPORTS STATISTICIANS ae employed y many pofessional spots teams, leagues, and news oganizations. They collect and analyze team and individual data on items such as scoing. CAREER LINK www.mcdougallittell.com INTERNET FOCUS ON CAREERS BASEBALL In Execises 35 and 36, use the following infomation. The Pythagoean Theoem of Baseall is a fomula fo appoximating a team s atio of wins to games played. Let R e the nume of uns the team scoes duing the season, A e the nume of uns allowed to opponents, W e the nume of wins, and T e the total nume of games played. Then the fomula W T R R + A appoximates the team s atio of wins to games played. Souce: Inside Spots 35. Solve the fomula fo W. 36. The 1998 New Yok Yankees scoed 965 uns and allowed 656. How many of its 16 games would you estimate the team won? FUNDRAISER In Execises 37 39, use the following infomation. You tennis team is having a fundaise. You ae going to help aise money y selling sun visos and aseall caps. 37. Wite an equation that epesents the total amount of money you aise. 38. How many vaiales ae in the equation? What does each epesent? 39. You team aises a total of $4480. Give thee possile cominations of sun visos and aseall caps that could have een sold if the pice of a sun viso is $3.00 and the pice of a aseall cap is $7.00. 40. GEOMETRY CONNECTION The fomula fo the aea of a cicle is A = π. The fomula fo the cicumfeence of a cicle is C = π. Wite a fomula fo the aea of a cicle in tems of its cicumfeence. 30 Chapte 1 Equations and Inequalities
Page 6 of 7 HOMEWORK HELP Visit ou We site www.mcdougallittell.com fo help with polem solving in Exs. 41 and 4. INTERNET 41. GEOMETRY CONNECTION The fomula fo the height h of an equilateal tiangle is 3 h = whee is the length of a side. Wite a fomula fo the aea of an equilateal tiangle in tems of the following. h a. the length of a side only. the height only Test Pepaation 4. GEOMETRY CONNECTION The suface aea S of a cylinde is given y the fomula S = πh + π. The height h of the cylinde shown at the ight is 5 moe than 3 times its adius. a. Wite a fomula fo the suface aea of the cylinde in tems of its adius.. Find the suface aea of the cylinde fo = 3, 4, and 6. QUANTITATIVE COMPARISON In Execises 43 and 44, choose the statement that is tue aout the given quantities. A The quantity in column A is geate. B The quantity in column B is geate. C The two quantities ae equal. D The elationship cannot e detemined fom the given infomation. h Column A Column B Skills Review Fo help with the Pythagoean theoem, see p. 917. 43. V = wh 7 cm V = wh 5 cm 7 cm 4 cm 3 cm 3 cm 44. V = π h V= π h 4 in. 6 in. 6 in. 4 in. 1.4 Rewiting Equations and Fomulas 31
Page 7 of 7 Challenge 45. FUEL EFFICIENCY The moe aeodynamic a vehicle is, the less fuel the vehicle s engine must use to ovecome ai esistance. To design vehicles that ae as fuel efficient as possile, automotive enginees use the fomula R = 0.0056 ª D C ª F A ª s EXTRA CHALLENGE www.mcdougallittell.com whee R is the ai esistance (in pounds), D C is the dag coefficient, F A is the fontal aea of the vehicle (in squae feet), and s is the speed of the vehicle (in miles pe hou). The fomula assumes that thee is no wind. a. Rewite the fomula to find the dag coefficient in tems of the othe vaiales.. Find the dag coefficient of a ca when the ai esistance is 50 pounds, the fontal aea is 5 squae feet, and the speed of the ca is 45 miles pe hou. MIXED REVIEW WRITING EXPRESSIONS Wite an expession to answe the question. (Skills Review, p. 99) 46. You uy x ithday cads fo $1.85 each. How much do you spend? 47. You have $30 and spend x dollas. How much money do you have left? 48. You dive 55 miles pe hou fo x hous. How many miles do you dive? 49. You have $50 in you ank account and you deposit x dollas. How much money do you now have in you account? 50. You spend $4 on x music cassettes. How much does each cassette cost? 51. A cetain all eaing weighs ounces. A ox contains x all eaings. What is the total weight of the all eaings? UNIT ANALYSIS Give the answe with the appopiate unit of measue. (Review 1.1) 7 metes 1 minute 168 hous 1 week 5. (60 minutes) 53. (5 weeks) 54. 4 1 4 feet + 7 3 4 feet 55. 13 1 4 lites º 8 7 lites 8 3 yads 15 dollas 56. (1 seconds) º 10 yads 57. 1 second (8 hous) + 45 dollas 1 hou SOLVING EQUATIONS Solve the equation. Check you solution. (Review 1.3) 58. 3d + 16 = d º 4 59. 5 º x = 3 + x 60. 10(y º 1) = y + 4 61. p º 16 + 4 = 4( º p) 6. º10x = 5x + 5 63. 1z = 4z º 56 64. 3 x º 7 = 1 65. º 3 x + 19 = º11 4 66. 1 4 x + 3 8 = 1 5 º 1 5 x 67. 5 4 x º 3 4 = 5 6 x + 1 3 Chapte 1 Equations and Inequalities