Systems of Equations and Inequalities

Transcription

1 CHAPTER CONNECTIONS Systms of Equations and Inqualitis CHAPTER OUTLINE.1 Linar Systms in Two Variabls with Applications 04. Linar Systms in Thr Variabls with Applications 16.3 Nonlinar Systms of Equations and Inqualitis 9.4 Systms of Inqualitis and Linar Programming 36 Th disposal of hazardous wast is a growing concrn for today s communitis, and with many budgts strtchd to th braking point, thr is a cost/bnfit analysis involvd. On major haulr uss trucks with a carrying capacity of 800 ft 3, and can transport at most 10 tons. A full containr of liquid wast wighs 800 lb and has a volum of 0 ft 3, whil a full containr of solid wast wighs 600 lb and has a volum of 30 ft 3. If th haulr maks $300 for disposing of liquid wast and $400 for disposing of solid wast, what is th maximum rvnu that can b gnratd pr truck? Chaptr outlins a systmatic procss for answring this qustion. This application appars as Exrcis 8 in Sction.4. Chck out ths othr ral-world connctions: Appropriat Masurmnts in Dittics (Sction.1, Exrcis 64) Allocating Winnings to Diffrnt Invstmnts (Sction., Exrcis 4) Minimizing Shipping Costs (Sction.4, Exrcis 61) Markt Pricing for Organic Produc (Sction.3, Exrcis 6) 03

2 .1 Linar Systms in Two Variabls with Applications Larning Objctivs In Sction.1 you will larn how to: A. Vrify ordrd pair solutions B. Solv linar systms by graphing C. Solv linar systms by substitution D. Solv linar systms by limination E. Rcogniz inconsistnt systms and dpndnt systms F. Us a systm of quations to modl and solv applications Childrn a c 3100 (10, 30) Adults a c 00 In arlir chaptrs, w usd linar quations in two variabls to modl a numbr of ralworld situations. Graphing ths quations gav us a visual imag of how th variabls wr rlatd, and hlpd us bttr undrstand this rlationship. In many applications, two diffrnt masurs of th indpndnt variabl must b considrd simultanously, lading to a systm of two linar quations in two unknowns. Hr, a graphical prsntation onc again supports a bttr undrstanding, as w xplor systms and thir many applications. A. Solutions to a Systm of Equations A systm of quations is a st of two or mor quations for which a common solution is sought. Systms ar widly usd to modl and solv applications whn th information givn nabls th rlationship btwn variabls to b statd in diffrnt ways. For xampl, considr an amusmnt park that brought in $3100 in rvnu by charging $9.00 for adults and $.00 for childrn, whil slling 00 tickts. Using a for adult and c for childrn, w could writ on quation modling th numbr of tickts sold: a c 00, and a scond modling th amount of rvnu brought in: 9a c To show that w r considring both quations simultanously, a larg lft brac is usd and th rsult is calld a systm of two quations in two variabls: a c 00 9a c 3100 numbr of tickts amount of rvnu W not that both quations ar linar and will hav diffrnt slop valus, so thir graphs must intrsct at som point. Sinc vry point on a lin satisfis th quation of that lin, this point of intrsction must satisfy both quations simultanously and is th solution to th systm. Th figur that accompanis Exampl 1 shows th point of intrscion for this systm is (10, 30). EXAMPLE 1 Vrifying Solutions to a Systm A. You v just larnd how to vrify ordrd pair solutions a c 00 Vrify that (10, 30) is a solution to. 9a c 3100 Solution Substitut th 10 for a and 30 for c in ach quation. a c first quation 9a c scond quation Sinc (10, 30) satisfis both quations, it is th solution to th systm and w find th park sold 10 adult tickts and 30 tickts for childrn. B. Solving Systms Graphically Now try Exrciss 7 through 18 To solv a systm of quations mans w apply various mthods in an attmpt to find ordrd pair solutions. As Exampl 1 suggsts, on mthod for finding solutions is to graph th systm. Any mthod for graphing th lins can b mployd, but to kp important concpts frsh, th slop-intrcpt mthod is usd hr. 04 -

3 -3 Sction.1 Linar Systms in Two Variabls with Applications 0 EXAMPLE Solving a Systm Graphically 4x 3y 9 Solv th systm by graphing: x y. Solution First writ ach quation in slop-intrcpt form (solv for y): B. You v just larnd how to solv linar systms by graphing y 4 4x 3y 9 x y S 3 x 3 y x For th first lin, with y-intrcpt 10, 3. y Th scond quation yilds x 1 with 10, as th y-intrcpt. Both ar thn graphd on th grid as shown. Th point of intrsction appars to b (3, 1), and chcking this point in both quations givs 4x 3y y x 4 3 substitut 3 for x and 1 for y x y This vrifis that (3, 1) is th solution to th systm. 4 y 3 x 3 (0, 3) y y x (0, ) (3, 1) x C. Solving Systms by Substitution Now try Exrciss 19 through Whil a graphical approach bst illustrats why th solution must b an ordrd pair, it dos hav on obvious drawback nonintgr solutions ar difficult to spot. Th 4x y 4 ordrd pair 1, 1 is th solution to but this would b difficult to y x, pinpoint as a prcis location on a hand-drawn graph. To ovrcom this limitation, w nxt considr a mthod known as substitution. Th mthod involvs convrting a systm of two quations in two variabls into a singl quation in on variabl by using 4x y 4 an appropriat substitution. For th scond quation says y is two mor y x, than x. W rason that all points on this lin ar rlatd this way, including th point whr this lin intrscts th othr. For this rason, w can substitut x for yin th first quation, obtaining a singl quation in x. EXAMPLE 3 Solving a Systm Using Substitution 4x y 4 Solv using substitution: y x. Solution Sinc y x, w can rplac y with x in th first quation. 4x y 4 4x 1x 4 x 4 first quation substitut x for y simplify x rsult Th x-coordinat is. To find th y-coordinat, substitut for x into ithr of th original quations. Substituting in th scond quation givs y x scond quation 1 substitut for x 1 10, 10 1 Th solution to th systm is 1, 1. Vrify by substituting for x and for y into both quations. Now try Exrciss 3 through 3 1

4 06 CHAPTER Systms of Equations and Inqualitis -4 If nithr quation allows an immdiat substitution, w first solv for on of th variabls, ithr x or y, and thn substitut. Th mthod is summarizd hr, and can actually b usd with ithr lik variabls or lik variabl xprssions. S Exrciss 7 to 60. C. You v just larnd how to solv linar systms by substitution Solving Systms Using Substitution 1. Solv on of th quations for x in trms of y or y in trms of x.. Substitut for th appropriat variabl in th othr quation and solv for th variabl that rmains. 3. Substitut th valu from stp into ithr of th original quations and solv for th othr unknown. 4. Writ th answr as an ordrd pair and chck th solution in both original quations. D. Solving Systms Using Elimination x y 13 Now considr th systm whr solving for any on of th variabls x 3y 7, will rsult in fractional valus. Th substitution mthod can still b usd, but oftn th limination mthod is mor fficint. Th mthod taks its nam from what happns whn you add crtain quations in a systm (by adding th lik trms from ach). If th cofficints of ithr x or y ar additiv invrss thy sum to zro and ar liminatd. For th systm shown, adding th quations producs y 6, giving y 3, thn x 1 using back-substitution (vrify). Whn nithr variabl trm mts this condition, w can multiply on or both quations by a nonzro constant to match up th cofficints, so an limination will tak plac. In doing so, w crat an quivalnt systm of quations, maning on 7x 4y 16 that has th sam solution as th original systm. For, multiplying 3x y 6 7x 4y 16 th scond quation by producs, giving x 4 aftr adding 6x 4y 1 th quations. Not th thr systms producd ar quivalnt, and hav th solution (4, 3) ( y 3 was found using back-substitution). 7x 4y 16 7x 4y x y 6 6x 4y 1 In summary, 7x 4y 16 x 4 Oprations that Produc an Equivalnt Systm 1. Changing th ordr of th quations.. Rplacing an quation by a nonzro constant multipl of that quation. 3. Rplacing an quation with th sum of two quations from th systm. Bfor bginning a solution using limination, chck to mak sur th quations ar writtn in th standard form Ax By C, so that lik trms will appar abov/blow ach othr. Throughout this chaptr, w will us R1 to rprsnt th quation in row 1 of th systm, R to rprsnt th quation in row, and so on. Ths dsignations ar usd to hlp dscrib and documnt th stps bing usd to solv a systm, as in Exampl 4 whr R1 R indicats th first quation has bn multiplid by two, with th rsult addd to th scond quation.

5 - Sction.1 Linar Systms in Two Variabls with Applications 07 EXAMPLE 4 Solving a Systm by Elimination x 3y 7 Solv using limination: 6y x 4 Solution Th scond quation is not in standard form, so w r-writ th systm as x 3y 7. If w add th quations now, w would gt 7x 3y 11, with x 6y 4 nithr variabl liminatd. Howvr, if w multiply both sids of th first quation by, th y-cofficints will b additiv invrss. Th sum thn rsults in an quation with x as th only unknown. R1 4x 6y 14 { R x 6y 4 add sum 9x 0y 18 9x 18 WORTHY OF NOTE x solv for x As th limination mthod involvs adding two Substituting for x back into ithr of th original quations yilds y 1. Th quations, it is somtims ordrd pair solution is (, 1). Vrify using th original quations. rfrrd to as th addition mthod for solving systms. Now try Exrciss 33 through 38 Th limination mthod is summarizd hr. If ithr quation has fraction or dcimal cofficints, w can clar thm using an appropriat constant multiplir. Solving Systms Using Elimination 1. Writ ach quation in standard form: Ax By C.. Multiply on or both quations by a constant that will crat cofficints of x (or y) that ar additiv invrss. 3. Combin th two quations using vrtical addition and solv for th variabl that rmains. 4. Substitut th valu from stp 3 into ithr of th original quations and solv for th othr unknown.. Writ th answr as an ordrd pair and chck th solution in both original quations. EXAMPLE Solving a Systm Using Elimination D. You v just larnd how to solv linar systms by limination Solv using limination: 8x 3 4y x 3y 1. Solution Multiplying th first quation by 8(8R1) and th scond quation by 6(6R) will clar th fractions from ach. 8 8R1 6R 11 8x y x 6y x 6 11 S 3y 611 3x 4y 6 Th x-trms can now b liminatd if w us 3R1 1 R. { add 3R1 1x 18y 6 R 1x 0y 30 sum 0x y 4 y 1 solv for y Substituting y 1 in ithr of th original quations yilds x 14, and th solution is 1 14, 1. Vrify by substituting in both quations. Now try Exrciss 39 through 44

6 08 CHAPTER Systms of Equations and Inqualitis -6 CAUTION B sur to multiply all trms (on both sids) of th quation whn using a constant multiplir. Also, not that for Exampl, w could hav liminatd th y-trms using R1 with 3R. E. Inconsistnt and Dpndnt Systms A systm having at last on solution is calld a consistnt systm. As sn in Exampl, if th lins hav diffrnt slops, thy intrsct at a singl point and th systm has xactly on solution. Hr, th lins ar indpndnt of ach othr and th systm is calld an indpndnt systm. If th lins hav qual slops and th sam y-intrcpt, thy ar idntical or coincidnt lins. Sinc on is right atop th othr, thy intrsct at all points, and th systm has an infinit numbr of solutions. Hr, on lin dpnds on th othr and th systm is calld a dpndnt systm. Using substitution or limination on a dpndnt systm rsults in th limination of all variabl trms and lavs a statmnt that is always tru, such as 0 0 or som othr simpl idntity. EXAMPLE 6 Solving a Dpndnt Systm 3x 4y 1 Solv using limination:. 6x 4 8y Solution WORTHY OF NOTE Writing th systm in standard form givs 3x 4y 1. By applying R1, w can 6x 8y 4 liminat th variabl x: Whn writing th solution to R1 6x 8y 4 a dpndnt systm using a paramtr, th solution can R 6x 8y 4 add b writtn in many diffrnt sum 0x 0y 0 ways. For instanc, if w lt 0 0 tru statmnt p 4b for th first coordinat of th solution to Exampl 6, w hav 314b 3 3b 3 as 4 th scond coordinat, and th solution bcoms (4b, 3b 3) for any constant b. Although w didn t xpct it, both variabls wr liminatd and th final statmnt is tru This indicats th systm is dpndnt, which th graph vrifis (th lins ar coincidnt). Writing both quations in slop-intrcpt form shows thy rprsnt th sam lin. 3x 4y 1 6x 8y 4 variabls ar liminatd 4y 3x 1 8y 6x 4 3x 4y 1 y 3 4 x 3 μ y 3 4 x 3 Th solutions of a dpndnt systm ar oftn writtn in st notation as th st of ordrd pairs (x, y), whr y is a spcifid function of x. For Exampl 6 th solution would b 1x, y0y 3 4x 36. Using an ordrd pair with an arbitrary variabl, calld a paramtr, is also common: ap, 3p 3b. 4 Now try Exrciss 4 through 6 y (0, 3) (4, 0) 6x 4 8y x E. You v just larnd how to rcogniz inconsistnt systms and dpndnt systms Finally, if th lins hav qual slops and diffrnt y-intrcpts, thy ar paralll and th systm will hav no solution. A systm with no solutions is calld an inconsistnt systm. An inconsistnt systm producs an inconsistnt answr, such as 1 0 or som othr fals statmnt whn substitution or limination is applid. In othr words, all variabl trms ar onc again liminatd, but th rmaining statmnt is fals. A summary of th thr possibilitis is shown hr for arbitrary slop m and y-intrcpt (0, b).

7 -7 Sction.1 Linar Systms in Two Variabls with Applications 09 Indpndnt m 1 m y Dpndnt m 1 m, b 1 b y Inconsistnt m 1 m, b 1 b y x x x On point in common All points in common No points in common F. Systms and Modling In prvious chaptrs, w solvd numrous ral-world applications by writing all givn rlationships in trms of a singl variabl. Many situations ar asir to modl using a systm of quations with ach rlationship modld indpndntly using two variabls. W bgin hr with a mixtur application. Although thy appar in many diffrnt forms (coin problms, mtal alloys, invstmnts, mrchandising, and so on), mixtur problms all hav a similar thm. Gnrally on quation is rlatd to quantity (how much of ach itm is bing combind) and on quation is rlatd to valu (what is th valu of ach itm bing combind). WORTHY OF NOTE EXAMPLE 7 Solving a Mixtur Application As an stimation tool, not that if qual amounts of th 60% and 80% alloys wr usd (7 oz ach), th rsult would b a 70% alloy (halfway in btwn). Sinc a 7% alloy is ndd, mor of th 80% gold will b usd. A jwlr is commissiond to crat a pic of artwork that will wigh 14 oz and consist of 7% gold. Sh has on hand two alloys that ar 60% and 80% gold, rspctivly. How much of ach should sh us? Solution Lt x rprsnt ouncs of th 60% alloy and y rprsnt ouncs of th 80% alloy. Th first quation must b x y 14, sinc th pic of art must wigh xactly 14 oz (this is th quantity quation). Th x ouncs ar 60% gold, th y ouncs ar 80% gold, and th 14 oz will b 7% gold. This givs th valu quation: x y x 0.8y Th systm is (aftr claring dcimals). 6x 8y 10 Solving for y in th first quation givs y 14 x. Substituting 14 x for y in th scond quation givs 7 6x 8y 10 6x 8114 x 10 x x 7 scond quation substitut 14 x for y simplify solv for x Substituting for x in th first quation givs y 1. Sh should us 3. oz of th 60% alloy and 10. oz of th 80% alloy. Now try Exrciss 63 through 70 Systms of quations also play a significant rol in cost-basd pricing in th businss world. Th costs involvd in running a businss can broadly b undrstood as ithr a fixd cost k or a variabl cost v. Fixd costs might includ th monthly rnt paid for facilitis, which rmains th sam rgardlss of how many itms ar producd and sold. Variabl costs would includ th cost of matrials ndd to produc th itm, which dpnds on th numbr of itms mad. Th total cost can thn b modld by

8 10 CHAPTER Systms of Equations and Inqualitis -8 C1x vx k for x numbr of itms. Onc a slling pric p has bn dtrmind, th rvnu quation is simply R1x px (pric tims numbr of itms sold). W can now st up and solv a systm of quations that will dtrmin how many itms must b sold to brak vn, prforming what is calld a brak-vn analysis. EXAMPLE 8 Solving an Application of Systms: Brak-Evn Analysis In hom businsss that produc itms to sll on Ebay, fixd costs ar asily dtrmind by rnt and utilitis, and variabl costs by th pric of matrials ndd to produc th itm. Karn s hom businss maks larg, dcorativ candls for all occasions. Th cost of matrials is $3.0 pr candl, and hr rnt and utilitis avrag $900 pr month. If hr candls sll for $9.0, how many candls must b sold ach month to brak vn? Solution Lt x rprsnt th numbr of candls sold. Hr total cost is C1x 3.x 900 (variabl cost plus fixd cost), and projctd rvnu is R1x 9.x. This givs th C 1x 3.x 900 systm. To brak vn, Cost Rvnu which givs R1x 9.x WORTHY OF NOTE 9.x 3.x 900 6x 900 x 10 This brak-vn concpt can also b applid in studis of supply and dmand, as wll as in th dcision to buy a nw car or applianc that will nabl you to brak vn ovr tim du to nrgy and fficincy savings. Th analysis shows that Karn must sll 10 candls ach month to brak vn. Now try Exrciss 71 through 74 Our final xampl involvs an application of uniform motion ( distanc rat # tim), and xplors concpts of grat importanc to th navigation of ships and airplans. As a simpl illustration, if you v vr walkd at your normal rat r on th moving walkways at an airport, you likly noticd an incras in your total spd. This is bcaus th rsulting spd combins your walking rat r with th spd w of th walkway: total spd r w. If you walk in th opposit dirction of th walkway, your total spd is much slowr, as now total spd r w. This sam phnomnon is obsrvd whn an airplan is flying with or against th wind, or a ship is sailing with or against th currnt. EXAMPLE 9 Solving an Application of Systms Uniform Motion F. You v just larnd how to us a systm of quations to modl and solv applications An airplan flying du south from St. Louis, Missouri, to Baton Roug, Louisiana, uss a strong, stady tailwind to complt th trip in only. hr. On th rturn trip, th sam wind slows th flight and it taks 3 hr to gt back. If th flight distanc btwn ths citis is 91 km, what is th cruising spd of th airplan (spd with no wind)? How fast is th wind blowing? Solution Lt r rprsnt th rat of th plan and w th rat of th wind. Sinc D RT, th flight to Baton Roug can b modld by 91 1r w1., and th rturn flight 91.r.w R1 by 91 1r w13. This producs th systm. Using and 91 3r 3w. R r w givs th quivalnt systm, which is asily solvd using r w limination with R1 R r r R1 R divid by Th cruising spd of th plan (with no wind) is kph. Using shows th wind is blowing at 30.4 kph. r w 304 Now try Exrciss 7 through 78

9 -9 Sction.1 Linar Systms in Two Variabls with Applications 11 TECHNOLOGY HIGHLIGHT Solving Systms Graphically Whn usd with car, graphing calculators offr an accurat way to solv linar systms and to chck 4x y 4 solution(s) obtaind by hand. W ll illustrat using th systm from Exampl 3:, whr w found th solution was 1, 1 y x. Figur.1 1. Solv for y in both quations:. Entr th quations as 10 y 4x 4 y x Y 1 4x 4 Y x 3. Graph using ZOOM 6 4. Prss nd TRACE (CALC) Y 1 4x 4 ENTER ENTER ENTER to Y x hav th calculator comput th point of intrsction. 10 Th coordinats of th intrsction appar as dcimal fractions at th bottom of th scrn (Figur.1). In stp 4, Th first ENTER slcts Y 1, th scond ENTER slcts Y and th third ENTER bypasss th GUESS option (this option is most oftn usd if th graphs intrsct at mor than on point). Th calculator automatically rgistrs th x-coordinat as its most rcnt ntry, and from th hom scrn, convrting it to a standard fraction (using MATH 1: Frac ENTER ) shows x. You can also gt an approximat solution by tracing along ithr lin towards th point of intrsction using th TRACE ky and th lft or right arrows. Solv ach systm graphically, using a graphing calculator x y 7 x 3y 3 Exrcis 1: Exrcis : y x 1 6 8x 3y.1 EXERCISES CONCEPTS AND VOCABULARY Fill in th blank with th appropriat word or phras. Carfully rrad th sction if ndd. 1. Systms that hav no solution ar calld systms.. Systms having at last on solution ar calld systms. 3. If th lins in a systm intrsct at a singl point, th systm is said to b and. 4. If th lins in a systm ar coincidnt, th systm is rfrrd to as and.. Th givn systms ar quivalnt. How do w obtain th scond systm from th first? 3 x 1 y 3 0.x 0.4y 1 4x 3y 10 x 4y 10 x y 8 6. For which solution mthod would 3x 4y, b mor fficint, substitution or limination? Discuss/Explain why.

10 1 CHAPTER Systms of Equations and Inqualitis -10 DEVELOPING YOUR SKILLS Show th lins in ach systm would intrsct in a singl point by writing th quations in slop-intrcpt form An ordrd pair is a solution to an quation if it maks th quation tru. Givn th graph shown hr, dtrmin which quation(s) hav th indicatd point as a solution. If th point satisfis mor than on quation, writ th systm for which it is a solution. 9. A 10. B 11. C 1. D 13. E 14. F Substitut th x- and y-valus indicatd by th ordrd pair to dtrmin if it solvs th systm x 4y 4 4x 3y 1 0.3x 0.4y 0.x 0.y 4 3x y 11 13, x y 13; 3x 7y 4 1 6, 7x 8y 1; 8x 4y 17 a 7 1x 30y ; 8, 1 b 4x 1y 7 8x 1y 11; a1, 1 3 b Solv ach systm by graphing. If th coordinats do not appar to b intgrs, stimat th solution to th narst tnth (indicat that your solution is an stimat). 3x y x y 9 x y x 3y 1 Solv ach systm using substitution. Writ solutions as an ordrd pair. x y x y 6 3x y 6 x y 3x y 10 3x y x 3y 1 4x y 7 x y x y 6. y 3x x y 19 y 3 4x 1 F E x 3y 3 y y x A B x C D Idntify th quation and variabl that maks th substitution mthod asist to us. Thn solv th systm. 3x 4y x y x y x 1.4y 11.4 x 6y x y 6 Solv using limination. In som cass, th systm must first b writtn in standard form. x 4y x 4y 4x 3y y x 19 x 3y x y 1 0.x 0.4y y x m 0.1n m 0.08n g 0.3h g 0.h u 1 4v u 3v 11 Solv using any mthod and idntify th systm as consistnt, inconsistnt, or dpndnt. 4x y x 8y 13 0.y 0.3x x 0.4y 1 6x y x 1 y 11 10x 3y 1.. y 0.x 3x y 19 x 4y 3 0.8x y x 1.y 9.3 x y 8x y 6 x y 8 x y 6 y 3x 3x y 19 y x 4x 17 6y 0.x 0.3y x 0.4y x 1 3y x 1 y 3 3x y y 6x 9 1.x 0.4y 0.y 1.x 1 y 9x 3x 3y x 3y 4 x.y 7a m 3n 1 3. b 4. a b 14 m 6n 4

11 -11 Sction.1 Linar Systms in Two Variabls with Applications 13 4a 3b. 6. 6b a 7 3p q 4 9p 4q 3 Th substitution mthod can b usd for lik variabls or for lik xprssions. Solv th following systms, using th xprssion common to both quations (do not solv for x or y alon). x 4y x 1 4y 8x 3y 4 8x y 36 x 11y 1 6x y y 8x y 6x 4 WORKING WITH FORMULAS 61. Uniform motion with currnt: 1R CT 1 D 1 1R CT D Th formula shown can b usd to solv uniform motion problms involving a currnt, whr D rprsnts distanc travld, R is th rat of th objct with no currnt, C is th spd of th currnt, and T is th tim. Chan-Li rows 9 mi up rivr (against th currnt) in 3 hr. It only took him 1 hr to row mi downstram (with th currnt). How fast was th currnt? How fast can h row in still watr? 6. Fahrnhit and Clsius tmpraturs: y 9 x 3 F y 91x 3 C Many popl ar familiar with tmpratur masurmnt in dgrs Clsius and dgrs Fahrnhit, but fw raliz that th quations ar linar and thr is on tmpratur at which th two scals agr. Solv th systm using th mthod of your choic and find this tmpratur. APPLICATIONS Solv ach application by modling th situation with a linar systm. B sur to clarly indicat what ach variabl rprsnts. Mixtur 63. Thatr productions: At a rcnt production of A Comdy of Errors, th Community Thatr brought in a total of $30,49 in rvnu. If adult tickts wr $9 and childrn s tickts wr $6.0, how many tickts of ach typ wr sold if 3800 tickts in all wr sold? 64. Milk-fat rquirmnts: A ditician nds to mix 10 gal of milk that is 1 % milk fat for th day s rounds. H has som milk that is 4% milk fat and som that is 1 1 % milk fat. How much of ach should b usd? 6. Filling th family cars: Chrok just filld both of th family vhicls at a srvic station. Th total cost for 0 gal of rgular unladd and 17 gal of prmium unladd was $ Th prmium gas was $0.10 mor pr gallon than th rgular gas. Find th pric pr gallon for ach typ of gasolin. 66. Houshold clanrs: As a claning agnt, a solution that is 4% vingar is oftn usd. How much pur (100%) vingar and % vingar must b mixd to obtain 0 oz of a 4% solution? 67. Alumni contributions: A walthy alumnus donatd $10,000 to his alma matr. Th collg usd th funds to mak a loan to a scinc major at 7% intrst and a loan to a nursing studnt at 6% intrst. That yar th collg arnd $63 in intrst. How much was loand to ach studnt? 68. Invsting in bonds: A total of $1,000 is invstd in two municipal bonds, on paying 10.% and th othr 1% simpl intrst. Last yar th annual intrst arnd on th two invstmnts was $133. How much was invstd at ach rat? 69. Saving mony: Bryan has bn doing odd jobs around th hous, trying to arn nough mony to buy a nw Dirt-Surfr. H savs all quartrs and dims in his piggy bank, whil h placs all nickls and pnnis in a drawr to spnd. So far, h has coins in th piggy bank, worth a total of $4.00. How many of th coins ar quartrs? How many ar dims? 70. Coin invstmnts: In 1990, Molly attndd a coin auction and purchasd som rar Flowing Hair fifty-cnt pics, and a numbr of vry rar twocnt pics from th Civil War Era. If sh bought 47 coins with a fac valu of $10.06, how many of ach dnomination did sh buy?

12 14 CHAPTER Systms of Equations and Inqualitis Lawn srvic: Dav and his sons run a lawn srvic, which includs mowing, dging, trimming, and arating a lawn. His fixd cost includs insuranc, his salary, and monthly paymnts on quipmnt, and amounts to $4000/mo. Th variabl costs includ gas, oil, hourly wags for his mploys, and miscllanous xpnss, which run about $7 pr lawn. Th avrag charg for fullsrvic lawn car is $11 pr visit. Do a brakvn analysis to (a) dtrmin how many lawns Dav must srvic ach month to brak vn and (b) th rvnu rquird to brak vn. 7. Production of mini-microwav ovns: Du to high markt dmand, a manufacturr dcids to introduc a nw lin of mini-microwav ovns for prsonal and offic us. By using xisting factory spac and rtraining som mploys, fixd costs ar stimatd at $8400/mo. Th componnts to assmbl and tst ach microwav ar xpctd to run $4 pr unit. If markt rsarch shows consumrs ar willing to pay at last $69 for this product, find (a) how many units must b mad and sold ach month to brak vn and (b) th rvnu rquird to brak vn. In a markt conomy, th availability of goods is closly rlatd to th markt pric. Supplirs ar willing to produc mor of th itm at a highr pric (th supply), with consumrs willing to buy mor of th itm at a lowr pric (th dmand). This is calld th law of supply and dmand. Whn supply and dmand ar qual, both th buyr and sllr ar satisfid with th currnt pric and w hav markt quilibrium. 73. Farm commoditis: On ara whr th law of supply and dmand is clarly at work is farm commoditis. Both growrs and consumrs watch this rlationship closly, and us data collctd by govrnmnt agncis to track th rlationship and mak adjustmnts, as whn a farmr dcids to convrt a larg portion of hr farmland from corn to soybans to improv profits. Suppos that for x billion bushls of soybans, supply is modld by y 1.x 3, whr y is th currnt markt pric (in dollars pr bushl). Th rlatd dmand quation might b y.0x 1. (a) How many billion bushls will b supplid at a markt pric of $.40? What will th dmand b at this pric? Is supply lss than dmand? (b) How many billion bushls will b supplid at a markt pric of $7.0? What will th dmand b at this pric? Is dmand lss than supply? (c) To th narst cnt, at what pric dos th markt rach quilibrium? How many bushls ar bing supplid/dmandd? 74. Digital music: Markt rsarch has indicatd that by 010, sals of MP3 portabls will mushroom into a $70 billion dollar markt. With a markt this larg, comptition is oftn firc with supplirs fighting to arn and hold markt shars. For x million MP3 playrs sold, supply is modld by y 10.x, whr y is th currnt markt pric (in dollars). Th rlatd dmand quation might b y.0x 140. (a) How many million MP3 playrs will b supplid at a markt pric of $88? What will th dmand b at this pric? Is supply lss than dmand? (b) How many million MP3 playrs will b supplid at a markt pric of $114? What will th dmand b at this pric? Is dmand lss than supply? (c) To th narst cnt, at what pric dos th markt rach quilibrium? How many units ar bing supplid/dmandd? Uniform Motion 7. Canoing on a stram: On a rcnt camping trip, it took Molly and Sharon hr to row 4 mi upstram from th drop in point to th campsit. Aftr a lisurly wknd of camping, fishing, and rlaxation, thy rowd back downstram to th drop in point in just 30 min. Us this information to find (a) th spd of th currnt and (b) th spd Sharon and Molly would b rowing in still watr. 76. Taking a luxury cruis: A luxury ship is taking a Caribban cruis from Caracas, Vnzula, to just off th coast of Bliz City on th Yucatan Pninsula, a distanc of 143 mi. En rout thy ncountr th Caribban Currnt, which flows to th northwst, paralll to th coastlin. From Caracas to th Bliz coast, th trip took 70 hr. Aftr a fw days of fun in th sun, th ship lavs for Caracas, with th rturn trip taking 8 hr. Us

13 -13 Sction.1 Linar Systms in Two Variabls with Applications 1 this information to find (a) th spd of th Caribban Currnt and (b) th cruising spd of th ship. 77. Airport walkways: As part of an algbra fild trip, Jason taks his class to th airport to us thir moving walkways for a dmonstration. Th class masurs th longst walkway, which turns out to b 6 ft long. Using a stop watch, Jason shows it taks him just 3 sc to complt th walk going in th sam dirction as th walkway. Walking in a dirction opposit th walkway, it taks him 30 sc 10 tims as long! Th nxt day in class, Jason hands out a two-qustion quiz: (1) What was th spd of th walkway in ft pr scond? () What is my (Jason s) normal walking spd? Crat th answr ky for this quiz. 78. Racing pigons: Th Amrican Racing Pigon Union oftn sponsors opportunitis for ownrs to fly thir birds in frindly comptitions. During a rcnt comptition, Stv s birds wr libratd in Topka, Kansas, and hadd almost du north to thir loft in Sioux Falls, South Dakota, a distanc of 308 mi. During th flight, thy ncountrd a stady wind from th north and th trip took 4.4 hr. Th nxt month, Stv took his birds to a comptition in Grand Forks, North Dakota, with th birds hading almost du south to hom, also a distanc of 308 mi. This tim th birds wr aidd by th sam wind from th north, and th trip took only 3. hr. Us this information to (a) find th racing spd of Stv s birds and (b) find th spd of th wind. EXTENDING THE CONCEPT 83. Answr using obsrvations only no calculations. Is th givn systm consistnt/indpndnt, consistnt/dpndnt, or inconsistnt? y x Explain/Discuss your answr. y.01x Fdral incom tax rform has bn a hot political topic for many yars. Suppos tax plan A calls for a flat tax of 0% tax on all incom (no dductions or loophols). Tax plan B rquirs taxpayrs to pay Dscriptiv Translation 79. Important dats in U.S. history: If you sum th yar that th Dclaration of Indpndnc was signd and th yar that th Civil War ndd, you gt Thr ar 89 yr that sparat th two vnts. What yar was th Dclaration signd? What yar did th Civil War nd? 80. Architctual wondrs: Whn it was first constructd in 1889, th Eiffl Towr in Paris, Franc, was th tallst structur in th world. In 197, th CN Towr in Toronto, Canada, bcam th world s tallst structur. Th CN Towr is 13 ft lss than twic th hight of th Eiffl Towr, and th sum of thir hights is 799 ft. How tall is ach towr? 81. Pacific islands land ara: In th South Pacific, th island nations of Tahiti and Tonga hav a combind land ara of 69 mi. Tahiti s land ara is 11 mi mor than Tonga s. What is th land ara of ach island group? 8. Card gams: On a cold wintr night, in th lobby of a bautiful hotl in Sant F, Nw Mxico, Marc and Klay just barly bat John and Stv in a clos gam of Trumps. If th sum of th tam scors was 990 points, and thr was a 1-point margin of victory, what was th final scor? $000 plus 10% of all incom. For what incom lvl do both plans rquir th sam tax? 8. Suppos a crtain amount of mony was invstd at 6% pr yar, and anothr amount at 8.% pr yar, with a total rturn of $10. If th amounts invstd at ach rat wr switchd, th yarly incom would hav bn $137. To th narst whol dollar, how much was invstd at ach rat? MAINTAINING YOUR SKILLS 86. (.6) Givn th parnt function f 1x x, sktch th graph of F1x x (3.3) Us th RRT to writ th polynomial in compltly factord form: 3x 4 19x 3 1x 7x (4.4) Solv for x (roundd to th narst thousandth): x (3.1) Graph y x 6x 16 by complting th squar and stat th intrval whr f 1x 0.