Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w


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1 1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a vaiable A fomula is an equation that elates two o moe quantities, usually epesented by vaiables. Some common fomulas ae shown below. Quantity Fomula Meaning of vaiables Distance d t d distance, ate, t time Tempeatue F 9 C 1 3 F degees Fahenheit, C degees Celsius Aea of a tiangle A 1 bh A aea, b base, h height Aea of a ectangle A lw A aea, l length, w width READING The vaiables b 1 and b ae ead as b sub one and b sub two. The small loweed numbes ae called subscipts. Peimete of a ectangle P l 1 w P peimete, l length, w width Aea of a tapezoid A 1 (b 1 1 b )h A aea, b 1 one base, b othe base, h height Aea of a cicle A π A aea, adius Cicumfeence of a cicle C π C cicumfeence, adius To solve fo a vaiable means to ewite an equation as an equivalent equation in which the vaiable is on one side and does not appea on the othe side. E XAMPLE 1 Rewite a fomula with two vaiables Solve the fomula C p fo. Then find the adius of a cicle with a cicumfeence of 44 inches. STEP 1 Solve the fomula fo. C π Wite cicumfeence fomula. STEP C Divide each side by p. π Substitute the given value into the ewitten fomula. C π 44 π ø 7 c The adius of the cicle is about 7 inches. Substitute 44 fo C and simplify. 6 Chapte 1 Equations and Inequalities
2 GUIDED PRACTICE fo Example 1 1. Find the adius of a cicle with a cicumfeence of feet.. The fomula fo the distance d between opposite vetices of a egula hexagon is d a whee a is the distance Ï 3 between opposite sides. Solve the fomula fo a. Then find a when d 10 centimetes. d a E XAMPLE Rewite a fomula with thee vaiables Solve the fomula P l 1 w fo w. Then find the width of a ectangle with a length of 1 metes and a peimete of 41 metes. STEP 1 Solve the fomula fo w. STEP P l 1 w P l w Wite peimete fomula. Subtact l fom each side. P l w Divide each side by. Substitute the given values into the ewitten fomula. w w (1) Substitute 41 fo P and 1 fo l. Simplify. c The width of the ectangle is 8. metes. at classzone.com P 41 m 1 m w GUIDED PRACTICE fo Example 3. Solve the fomula P l 1 w fo l. Then find the length of a ectangle with a width of 7 inches and a peimete of 30 inches. 4. Solve the fomula A lw fo w. Then find the width of a ectangle with a length of 16 metes and an aea of 40 squae metes. Solve the fomula fo the vaiable in ed. Then use the given infomation to find the value of the vaiable.. A 1 bh 6. A 1 bh 7. A 1 (b 1 1 b )h b 1 h h h b b b Find h if b 1 m Find b if h 3 cm Find h if b 1 6 in., and A 84 m. and A 9 cm. b 8 in., and A 70 in. 1.4 Rewite Fomulas and Equations 7
3 REWRITING EQUATIONS The appoach you use to solve a fomula fo a vaiable can be applied to othe algebaic equations. E XAMPLE 3 Rewite a linea equation Solve 9x 4y 7 fo y. Then find the value of y when x. AVOID ERRORS When dividing each side of an equation by the same numbe, emembe to divide evey tem by the numbe. STEP 1 Solve the equation fo y. STEP 9x 4y 7 4y 7 9x Wite oiginal equation. Subtact 9x fom each side. y x Divide each side by 4. 4 Substitute the given value into the ewitten equation. y () Substitute fo x. 4 y y 13 Multiply. Simplify. CHECK 9x 4y 7 Wite oiginal equation. 9() 4(13) 0 7 Substitute fo x and 13 fo y. 7 7 checks. E XAMPLE 4 Rewite a nonlinea equation Solve y 1 xy 6 fo y. Then find the value of y when x 3. AVOID ERRORS If you ewite the equation as y 6 y x, then you have not solved fo y because y still appeas on both sides of the equation. STEP 1 Solve the equation fo y. y 1 xy 6 Wite oiginal equation. ( 1 x)y 6 Distibutive popety y 6 1 x Divide each side by ( 1 x). STEP Substitute the given value into the ewitten equation. y 6 1 (3) Substitute 3 fo x. y 6 Simplify. GUIDED PRACTICE fo Examples 3 and 4 Solve the equation fo y. Then find the value of y when x. 8. y 6x 7 9. y x x 1 y x 1 y xy x 13. 4y xy 8 8 Chapte 1 Equations and Inequalities
4 E XAMPLE Solve a multistep poblem MOVIE RENTAL A video stoe ents new movies fo one pice and olde movies fo a lowe pice, as shown at the ight. Wite an equation that epesents the stoe s monthly evenue. Solve the evenue equation fo the vaiable epesenting the numbe of new movies ented. The owne wants $1,000 in evenue pe month. How many new movies must be ented if the numbe of olde movies ented is 00? 1000? STEP 1 Wite a vebal model. Then wite an equation. Monthly evenue (dollas) Pice of new movies (dollas/movie) p Numbe of new movies (movies) 1 Pice of olde movies (dollas/movie) p Numbe of olde movies (movies) R p n p n An equation is R n 1 1 3n. STEP Solve the equation fo n 1. R n 1 1 3n R 3n n 1 Wite equation. Subtact 3n fom each side. R 3n n 1 Divide each side by. STEP 3 Calculate n 1 fo the given values of R and n. 1,000 3 p 00 If n 00, then n ,000 3 p 1000 If n 1000, then n c If 00 olde movies ae ented, then 100 new movies must be ented. If 1000 olde movies ae ented, then 1800 new movies must be ented. GUIDED PRACTICE fo Example 14. WHAT IF? In Example, how many new movies must be ented if the numbe of olde movies ented is 100? 1. WHAT IF? In Example, how many new movies must be ented if customes ent no olde movies at all? 16. Solve the equation in Step 1 of Example fo n. 1.4 Rewite Fomulas and Equations 9
5 1.4 EXERCISES SKILL PRACTICE HOMEWORK KEY WORKEDOUT SOLUTIONS on p. WS1 fo Exs. 3, 9, and 3 STANDARDIZED TEST PRACTICE Exs., 6, 1, 7, 36, and VOCABULARY Copy and complete: A(n)? is an equation that elates two o moe quantities.. WRITING What does it mean to solve fo a vaiable in an equation? EXAMPLES 1 and on pp. 6 7 fo Exs. 3 6 REWRITING FORMULAS Solve the fomula fo the indicated vaiable. Then use the given infomation to find the value of the vaiable. 3. Solve A lw fo l. Then find the length of a ectangle with a width of 0 millimetes and an aea of 0 squae millimetes. 4. Solve A 1 bh fo b. Then find the base of a tiangle with a height of 6 inches and an aea of 4 squae inches.. Solve A 1 (b 1 1 b )h fo h. Then find the height of a tapezoid with bases of lengths 10 centimetes and 1 centimetes and an aea of 7 squae centimetes. 6. MULTIPLE CHOICE What equation do you obtain when you solve the fomula A 1 (b 1 1 b )h fo b 1? A b 1 A h b B b 1 A h b C b 1 A b h D b 1 A h b EXAMPLE 3 on p. 8 fo Exs REWRITING EQUATIONS Solve the equation fo y. Then find the value of y fo the given value of x. 7. 3x 1 y 6; x y 1 x 4; x x 1 y 31; x x 1 4y 9; x x 6y 63; x 1. 10x 18y 84; x y 14x ; x 14. 9y 4x 30; x 8 1. MULTIPLE CHOICE What equation do you obtain when you solve the equation 4x y 0 fo y? A x 4 y 1 B y 4 x 1 4 C y 4 x 4 D y 4 x 0 ERROR ANALYSIS Descibe and coect the eo in solving the equation fo y x 1 y 17. 4y xy 9 y 7x 1 4y 9 1 xy y 7 x 1 y 9 1 xy 4 30 Chapte 1 Equations and Inequalities
6 GEOMETRY Solve the fomula fo the vaiable in ed. Then use the given infomation to find the value of the vaiable. Round to the neaest tenth. 18. Aea of a 19. Lateal suface aea 0. Volume of cicula ing of a tuncated cylinde an ellipsoid A πw S π(h 1 k) V 4 3 πabc w k h c b a Find if w 4 ft Find h if cm, Find c if a 4 in., and A 10 ft. k 3 cm, and S 0 cm. b 3 in., and V 60 in. 3 EXAMPLE 4 on p. 8 fo Exs. 1 6 REWRITING EQUATIONS Solve the equation fo y. Then find the value of y fo the given value of x. 1. xy 3x 40; x. 7x xy 18; x xy 8 16x; x y 1 6xy 30; x 6. y xy 1; x x 1 7y 1 xy 0; x 1 7. SHORT RESPONSE Conside the equation 1x 9y 7. To find the value of y when x, you can use two methods. Method 1 Solve the oiginal equation fo y and then substitute fo x. Method Substitute fo x and then solve the esulting equation fo y. Show the steps of the two methods. Which method is moe efficient if you need to find the value of y fo seveal values of x? Explain. REASONING Solve fo the indicated vaiable. 8. Solve xy x 1 y fo y. 9. Solve xyz x 1 y 1 z fo z. 30. Solve 1 x 1 1 y 1 fo y. 31. Solve 1 x 1 1 y 1 1 z 1 fo z. 3. CHALLENGE Wite a fomula giving the aea of a cicle in tems of its cicumfeence. PROBLEM SOLVING EXAMPLE on p. 9 fo Exs TREE DIAMETER You can estimate the diamete of a tee without boing though it by measuing its cicumfeence. Solve the fomula C πd fo d. Then find the diamete of an oak that has a cicumfeence of 113 inches. 34. DESIGN The fabic panels on a kite ae hombuses. A fomula fo the length of the long diagonal d is d s Ï 3 whee s is the length of a side. Solve the fomula fo s. Then find the value of s when d 1 inches. s s d s s 1.4 Rewite Fomulas and Equations 31
7 3. TEMPERATURE The fomula fo conveting tempeatues fom degees Celsius to degees Fahenheit is F 9 C 1 3. Solve the fomula fo C. Then find the tempeatue in degees Celsius that coesponds to 08F. 36. EXTENDED RESPONSE A quate mile unning tack is shaped as shown. The fomula fo the inside peimete is P π 1 x. a. Solve the peimete fomula fo. b. Fo a quate mile tack, P 440 yads. Find when x 7 yads, 100 yads, 10 yads, and 10 yads. c. What ae the geatest and least possible values of if P 440 yads? Explain how you found the values, and sketch the tack coesponding to each exteme value. x 37. MULTISTEP PROBLEM A tuxedo shop ents classic tuxedos fo $80 and designe tuxedos fo $10. Wite an equation that epesents the shop s evenue. Solve the equation fo the vaiable epesenting the numbe of designe tuxedos ented. The shop owne wants $60,000 in evenue duing pom season. How many designe tuxedos must be ented if the numbe of classic tuxedos ented is 600? 40? 300? 38. OPENENDED MATH The volume of a donutlike shape called a tous is given by the fomula V π R whee and R ae the adii shown and R. a. Solve the fomula fo R. R R b. A metal ing in the shape of a tous has a volume of 100 cubic centimetes. Choose thee possible values of, and find the coesponding values of R. l w 39. CHALLENGE A ectangula piece of pape with length l and width w can be olled to fom the lateal suface of a cylinde in two ways, assuming no ovelapping. Wite a fomula fo the volume of each cylinde in tems of l and w. w l MIXED REVIEW PREVIEW Pepae fo Lesson 1. in Exs Wite an expession to answe the question. (p. 984) 40. You have $0 in a bank account and deposit x dollas. What is you cuent balance? 41. You buy x CDs fo $1.99 each. How much do you spend? Evaluate the expession fo the given value of the vaiable. (p. 10) 4. 6j 1 8 when j k 4 when k g 8g p when g 1 4. m 3 1 m when m (n 1 7) 4 when n 47. (3p 17) 3 when p Solve the equation. Check you solution. (p. 18) 48. 4x x y y (4 1 z) 1. 9(p 1 3) 3p q Chapte 1 EXTRA Equations PRACTICE and Inequalities fo Lesson 1.4, p ONLINE QUIZ at classzone.com
8 MIXED REVIEW of Poblem Solving Lessons MULTISTEP PROBLEM Thee is a $0 annual membeship fee to join an uban ca ental sevice. Using a ca costs $8.0 pe hou. a. Wite a vebal model fo this situation. Then use the vebal model to wite an algebaic expession. b. How much will it cost to join the sevice and dive fo 0 hous? STATE TEST PRACTICE classzone.com. GRIDDED ANSWER You dive fom Chicago to St. Louis, a distance of 90 miles. You aveage speed is 60 miles pe hou. How many hous does the tip take? Round you answe to the neaest tenth of an hou. 6. OPENENDED Descibe a shopping situation that can be modeled by the equation 10x 1 9y 78.. MULTISTEP PROBLEM You ae attending a museum. You have $0 to spend. Admission to the museum is $1. Admission to each special exhibit inside the museum is $10. a. Wite an equation that can be used to find the numbe of special exhibits you can include in you visit. b. Solve the equation. Intepet you answe in tems of the poblem. 3. SHORT RESPONSE In hockey, each playe has a statistic called plus/minus, which is the diffeence between the numbe of goals scoed by the playe s team and the numbe of goals scoed by the othe team when the playe is on the ice. List the playes shown in ode fom least to geatest plus/minus. Whose plus/ minus scoe is best? Explain. Playe Plus/Minus Vincent Lecavalie 3 Dave Andeychuk 9 Ruslan Fedotenko 14 Matin St. Louis 3 Coy Saich Tim Taylo 4. SHORT RESPONSE You ae in chage of buying food fo a school picnic. You have $4 to spend on gound beef and chicken. Gound beef costs $1.80 pe pound and chicken costs $1.00 pe pound. Wite an equation epesenting the situation. You want to buy equal amounts of gound beef and chicken. How much of each can you buy? Show how you found you answe. 7. EXTENDED RESPONSE In one yea, the Bueau of Engaving and Pinting pinted $10 and $0 bills with a total value of $66,368,000. The total numbe of $10 and $0 bills was 3,77,600. Numbe Value $10 bills x 10x $0 bills?? Total 3,77,600 66,368,000 a. Copy and complete the table. b. Wite and solve an equation to find how many $10 bills and how many $0 bills wee pinted. c. Compae the total value of the $10 bills pinted with the total value of the $0 bills pinted. 8. OPENENDED You have two summe jobs. You mow lawns fo $0 pe lawn. You also wok at a estauant fo $7.0 pe hou. Wite an equation fo the total amount of money you ean. Then find thee diffeent ways to ean $300 duing one week. 9. GRIDDED ANSWER The liopleuodon, a swimming dinosau fom the Late Juassic peiod, could gow to metes in length. Use the fact that 1 in..4 cm to convet the length to feet. Round you answe to the neaest foot. 10. GRIDDED ANSWER The fomula fo the volume of a cone is V 1 Bh. Find h (in centimetes) if 3 V 176 cm 3 and B 40 cm. Mixed Review of Poblem Solving 33
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