ࡱ>  Tf[ ]]bjbj 4(ΐΐ]Unn4trg.g.g.g.B/B/B/+------$ nQ6B/B/66Qg.g.fQ8Q8Q86:g.g.+Q86+Q8Q8B/g.>d`:7f|0y*0)8(00tB/1Q83|05B/B/B/QQQ8B/B/B/66660B/B/B/B/B/B/B/B/B/n w: Systems of Linear Equations (3 EMBED Equation.3 3) A system of linear equations can consist of two equations with two variables, three equations with three variables, four equations with four variables etc. To solve a system of three equations with three unknowns, means to find one value for each variable that will satisfy each of the equations. This process is usually done algebraically by using the process of elimination. The steps involved in solving a 3 EMBED Equation.3 3 system of linear equations algebraically are: Set up two pair of two equations and eliminate the same variable from each pair. There is now a 2 EMBED Equation.3 2 system of linear equations to solve. The equations can be solved using any of the known methods used to solve a 2 EMBED Equation.3 2 system. When the solution for the 2 EMBED Equation.3 2 system of linear equations has been determined, these numbers must be substituted into one of the original equations of the given system of linear equations. This new equation must be solved in order to find the value of the final variable. Example1: Solve the following system of linear equations algebraically:  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  These are the two pair of two equations. In each pair, the variable y has been eliminated. There is now a 2 EMBED Equation.3 2 system of linear equations which must be solved.  EMBED Equation.3   EMBED Equation.3  This 2 EMBED Equation.3 2 system is being solved by elimination.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The solution is (x, y, z) = (1,-2,-1). The answers for x and z were substituted into one of the equations of the system of linear equations. It makes no difference which equation you substitute the values into since the solution will satisfy all of the equations. A 3 EMBED Equation.3 3 system of linear equations can also be solved by using the inverse matrix method. Because this process is very long and cumbersome to do algebraically, it is normally done using a graphing calculator. Solving a 3 EMBED Equation.3 3 system using technology will be presented in the Appendix. Example 2: Solve the following system of linear equations algebraically:  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  The solution is (a, b, c) = (-2, -1, -3). Example 3: Solve the following system of linear equations algebraically:  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  Expressed as a decimal  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The solution is (x, y, z)  EMBED Equation.3  (0.17, 1.71, 3.47). The above example has a solution in which the values are in decimal form. These answers are approximate values and this must be considered when the values are replaced for the variables in the system of equations. The results will be approximate and not exact. Example 4: Solve the following system of linear equations algebraically:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The solution is (r, s, t) = (3,-2, 2). Three by three systems of linear equations are also used to solve real-life problems. The given problem is expressed as a system of linear equations and then solved to determine the value of the variables. Sometimes, the system will consist of three equations but not every equation will have three variables. Example three is one such problem. Example 1: Solve the following problem using your knowledge of systems of linear equations. Jesse, Maria and Charles went to the local craft store to purchase supplies for making decorations for the upcoming dance at the high school. Jesse purchased three sheets of craft paper, four boxes of markers and five glue sticks. His bill, before taxes was $24.40. Maria spent $30.40 when she bought six sheets of craft paper, five boxes of markers and two glue sticks. Charles, purchases totaled $13.40 when he bought three sheets of craft paper, two boxes of markers and one glue stick. Determine the unit cost of each item. Let p represent the number of sheets of craft paper. Let m represent the number of boxes of markers. Let g represent the number of glue sticks. Express the problem as a system of linear equations:  EMBED Equation.3  Solve the system of linear equations to determine the unit cost of each item.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The unit cost of each item is: 1 sheet of craft paper = $1.75 1 box of markers = $3.60 1 glue stick = $0.95 Example 2: Solve the following problem using your knowledge of systems of linear equations. Rafael, an exchange student from Brazil, made phone calls within Canada, to the United States, and to Brazil. The rates per minute for these calls vary for the different countries. Use the information in the following table to determine the rates. MonthTime within Canada (min)Time to the U.S. (min)Time to Brazil (min)Charges ($)September90120180$252.00October70100120$184.00November50110150$206.00 Let c represent the rate for calls within Canada. Let u represent the rate for calls to the United States. Let b represent the rate for calls to Brazil. Express the problem as a system of linear equations:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The cost of minutes within Canada is $0.40/min. The cost of minutes to the United States is $0.60/min. The cost of minutes to Brazil is $0.80/min. Example 3: Solve the following problem using your knowledge of systems of linear equations. Calculate the number of minutes that Carlos called within Canada, to the United States, and to Mexico during the month of December. The charges are 28/min within Canada, 30/min to the U.S., and 84/min to Mexico if the following conditions applied: His total bill for the month was $90.96 He talked twice as long to Mexico as he did to the U.S. The total number of minutes spent talking within Canada and to Mexico was 122. Let c represent the number of minutes within Canada Let u represent the number of minutes to the United States Let m represent the number of minutes to Mexico Express the problem as a system of linear equations:  EMBED Equation.3  Solve  EMBED Equation.3  for c  EMBED Equation.3   EMBED Equation.3  Replace m with 2u in  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Now replace c and m in the first equation to get an equation in terms of the variable u.  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Carlos talked for 42 minutes within Canada, to the U.S. for 40 minutes and to Mexico for 80 minutes. Example 4: Solve the following problem using your knowledge of systems of linear equations. Tracy, Danielle and Sherri bought snacks for a girls sleepover. They each bought the items shown in the following table at the local convenience store: Number of bags of potato chipsNumber of litres of popNumber of chocolate barsCost ($)44621.00321020.8823413.17 Calculate the unit price of each snack purchased by the girls. Let c represent the unit cost of the potato chips. Let p represent the unit cost of the pop. Let b represent the unit cost a chocolate bar. Express the problem as a system of linear equations:  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The price of one bag of potato chips is $1.98. The price of one litre of pop is $1.47 The price of one chocolate bar is $1.20 Exercises: 1. Solve the following systems of linear equations algebraically: a)  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  d) EMBED Equation.3  2. A local computer company sells three types of laptop computers to three nearby stores. The number of laptops ordered by each store and the amount owing to the company for the order is shown in the following table: StoreLaptop ALaptop BLaptop CAmount Owing($)Wal-Mart108621 200Sears79518 700Target84313 000 Write a system of equations to represent the above information and determine the unit price of each type of laptop computer. 3. Cory, Josh and Dan went shopping for Halloween treats. Cory bought 3 chocolate pumpkins, 4 masks and 8 candy witches. He spent $36.65. Josh bought 5 chocolate pumpkins, 3 masks and 10 candy witches. He spent $37.50. Dan bought 4 chocolate pumpkins, 5 masks and 6 candy witches. He spent $43.45. Write a system of equations to represent this problem and algebraically calculate the unit price of each item purchased. 4. Janet, Larry and Sam bought decorations to decorate the clubhouse for a Christmas party. The number of items bought by each person is given in the table along with the total cost of each purchase. Write a system of equations to represent this problem and algebraically calculate the unit price of each item. Each item was bought at the same store. Name of the shopperNumber of rolls of garlandNumber of wreathsNumber of poinsettiasCostJanet242$49.50Larry324$57.75Sam331$38.50 Solutions: 1.a)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The solution is (a, b, c) = (4, 2, -3) b)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The solution is (p, q, r) = (3, -2, 2) c)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The solution is (x, y, z) = (2, -2, 1) d)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  The solution is (c, d, e) = (0, -2, -3) 2. Let a represent the price of Laptop A. Let b represent the price of Laptop B. Let c represent the price of Laptop C.  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Laptop A costs $800.00. Laptop B costs $900.00. Laptop C costs $1000.00 3. Let p represent the unit price of the chocolate pumpkin. Let m represent the unit price of the mask. Let w represent the unit price of the candy witch.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Each chocolate pumpkin costs $2.55. Each mask costs $5.75. Each candy witch costs $.75. 4. Let g represent the cost of the garland. Let w represent the cost of a wreath. Let p represent the cost of a poinsettia.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The cost of each roll of garland is $3.75. The cost of each wreath is $6.25. 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Sc\%,IT:Rx=N Aa=3~Gݕd{XԕQJwJd\WӜ9F%!&qꔏv?/ '̡В|CKLɢ՝ :[#`CᶃQo<c1J=?ŅXvZzӔ+&r1EJf'J;`X*_~?48Dd c b n c $AV? ?3"`?m2Zm5EWHg즶1`!Zm5EWHg즶1Z@xڕO@߻_a"\Rhbb?0 HNI''wdu1ls}߻B@A@s-XȆáRw.²ւf`Hb9q랏8lV {HNiSP\Ol\.yGh>*=\sZPQJ0\nG)TJ2LOAZH~Qi}Y ]Qi\ {[(.m|%bSr W_g-{z:ٷ#9TB$M?5 we^%/OցQ( Q=WM-/<*¸sƑ^sP*A|(У<{AG8Ч8(LWDd @b o c $AW? ?3"`?n2 ([?\  }V`!u ([?\  j  & Cxcdd``6ed``baV d,FYzP1n:&B@?b u 00 UXRY7S?&,e`abM-VK-WMcX|"+|-JfF\ v`'@Fq`" y38߀$oh=  W&00|]Q{@$CompObj]fObjInfo_Equation Native `_1262285851F+Q`:+Q`:{h"d 6p+5m+2g=30.403p+2m+g=13.40 FMicrosoft Equation 3.0 DS Equation Equation.39q <9 !Ole cCompObjdfObjInfofEquation Native g)_1262285893 F+Q`:+Q`:Ole hCompObjifObjInfok FMicrosoft Equation 3.0 DS Equation Equation.39q T 6p+5m+2g=30.40"2(3p+2m+g=13.40)Equation Native l_1262285942F+Q`:+Q`:Ole oCompObjpf FMicrosoft Equation 3.0 DS Equation Equation.39q <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39qObjInforEquation Native s)_1262285948F+Q`:+Q`:Ole tCompObjufObjInfowEquation Native x_1262286046 F+Q`:+Q`:PE 6p+5m+2g=30.40"6p"4m"2g="26.80 FMicrosoft Equation 3.0 DS Equation Equation.39q!<T m=3.60Ole {CompObj |fObjInfo ~Equation Native =_1262286098F+Q`:+Q`:Ole CompObj fObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qxE "12p"6m="42.60"12p"6(3.60)="42.60"12p"21.60="42.60"12Equation Native _1262286290F+Q`:+Q`:Ole CompObjfp"21.60+21.60="42.60+21.60"12p="21"12"12p="21"12p=1.75 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _1262288380F+Q`:+Q`:Ole 8G 3p+2m+g=13.403(1.75)+2(3.60)+g=13.405.25+7.20+g=13.4012.45+g=13.4012.45"12.45+g=13.40"12.45g=.95CompObjfObjInfoEquation Native )_1262288489F+Q`:+Q`: FMicrosoft Equation 3.0 DS Equation Equation.39q $4; 90c+120u+180b=$252.0070c+100u+120b=$184.0050c+110u+150b=$206.00Ole CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q(k 90c+120u+180b=252.0070c+100u+120b=184.00_1262288538p"F+Q`:+Q`:Ole CompObj!#fObjInfo$ FMicrosoft Equation 3.0 DS Equation Equation.39q <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native )_1262288546'F+Q`:+Q`:Ole CompObj&(fObjInfo)Equation Native _1262288612%/,F+Q`:+Q`:Ole      A !"#$%&'()*+,-/.0213457689:<;=>?B@CDEFGIHJKLNMOQPRSVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~w 2(90c+120u+180b=252.00)"3(70c+100u+120b=184.00) FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj+-fObjInfo.Equation Native )_12622886191F+Q`:+Q`: <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39q` 180c+240u+360b=504.00"Ole CompObj02fObjInfo3Equation Native 210c"300u"360b="552.00 FMicrosoft Equation 3.0 DS Equation Equation.39qE$ "30c"60u="48.00_1262288712*>6F+Q`:+Q`:Ole CompObj57fObjInfo8Equation Native a_1262288755;F+Q`:+Q`:Ole CompObj:<f FMicrosoft Equation 3.0 DS Equation Equation.39q 70c+100u+120b=184.0050c+110u+150b=206.00ObjInfo=Equation Native _12622888059C@F+Q`:+Q`:Ole  FMicrosoft Equation 3.0 DS Equation Equation.39q <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj?AfObjInfoBEquation Native )_1262288810EF+Q`:+Q`:Ole CompObjDFfObjInfoGEquation Native `pD "5(70c+100u+120b=184.00)4(50c+110u+150b=206.00) FMicrosoft Equation 3.0 DS Equation Equation.39q_12622889004\JF+Q`:+Q`:Ole CompObjIKfObjInfoL <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39qxl "350c"500u"600b="920.0Equation Native )_1262288904OF+Q`:S`:Ole CompObjNPfObjInfoQEquation Native _1262288978MWTFS`:S`:Ole 0200c+440u+600b=824.00 FMicrosoft Equation 3.0 DS Equation Equation.39qI`pk "150c"60u="96.00CompObjSUfObjInfoVEquation Native e_1262289088YFS`:S`:Ole CompObjXZfObjInfo[Equation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q`( "30c"60u="48.00"150c"60u="96.00_1262289154Rf^FS`:S`:Ole CompObj]_fObjInfo` FMicrosoft Equation 3.0 DS Equation Equation.39q <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native )_1262289157cFS`:S`:Ole CompObjbdfObjInfoeEquation Native _1262289200akhFS`:S`:Ole    !"#&+./01234567:=>ADGLORUX]`cfghijmruvwx{~$o "1("30c"60u="48.00)"150c"60u="96.00 FMicrosoft Equation 3.0 DS Equation Equation.39q <9 !CompObjgifObjInfojEquation Native )_1262289203zmFS`:S`:Ole CompObjln fObjInfoo Equation Native   FMicrosoft Equation 3.0 DS Equation Equation.39qM\ 30c+60u=48.00"150c"60u="96.00_1262289602HrFS`:S`:Ole CompObjqsfObjInfot FMicrosoft Equation 3.0 DS Equation Equation.39q:K "120c="48.00"120"120c="48.00"120c=.40Equation Native _1262289840wFS`:S`:Ole CompObjvxf FMicrosoft Equation 3.0 DS Equation Equation.39q@B "30c"60u="48.00"30(.40)"60u="48.00"12.00"60u="48.00"12.00+12.00"60u="ObjInfoyEquation Native  _1262289490|FS`:S`:Ole $48.00+12.00"60u="36.00"60"60u="36.00"60u=.60 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj{}%fObjInfo~'Equation Native (#_1262289851uFS`:S`:9K  FMicrosoft Equation 3.0 DS Equation Equation.39q}0BK 70c+100u+120b=184.0070(.40)+100(.60)+1Ole )CompObj*fObjInfo,Equation Native -20b=184.0028.00+60.00+120b=184.0088.00+120b=184.0088.00"88.00+120b=184.00"88.00120b=96.00120120b=96.00120b=.80_1262291741FS`:S`:Ole 8CompObj9fObjInfo; FMicrosoft Equation 3.0 DS Equation Equation.39q9K .28c+.30u+.84m=$90.962u=mc+m=122Equation Native <_1262291773FS`:S`:Ole ?CompObj@f FMicrosoft Equation 3.0 DS Equation Equation.39q%( c+m=122 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoBEquation Native CA_1262291818FS`:S`:Ole ECompObjFfObjInfoHEquation Native I)_1262291822FS`:S`: <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39q%) c=122"mOle JCompObjKfObjInfoMEquation Native NA_1262291896FS`:S`:Ole PCompObjQfObjInfoS FMicrosoft Equation 3.0 DS Equation Equation.39q%) c=122"m FMicrosoft Equation 3.0 DS EqEquation Native TA_1262291915FS`:S`:Ole VCompObjWfuation Equation.39q <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39q)H'lO c=122"2ObjInfoYEquation Native Z)_1262291918FS`:S`:Ole [CompObj\fObjInfo^Equation Native _E_1262292375FS`:S`:u FMicrosoft Equation 3.0 DS Equation Equation.39q;pt .28(122"2u)+.30u+.84(2u)=90.9634.16".5Ole aCompObjbfObjInfodEquation Native eW6u+.30u+1.68u=90.9634.16+1.42u=90.96 FMicrosoft Equation 3.0 DS Equation Equation.39q <9 !_1262292235FS`:S`:Ole kCompObjlfObjInfonEquation Native o)_1262292242FS`:S`:Ole pCompObjqf FMicrosoft Equation 3.0 DS Equation Equation.39q!Xi 34.16"34.16+1.42u=90.96"34.161.42u=56.801.421.42u=ObjInfosEquation Native t=_1262292402FS`:S`:Ole y56.801.42u=40 FMicrosoft Equation 3.0 DS Equation Equation.39qMt 2u=m2(40)=m80=mCompObjzfObjInfo|Equation Native }i_1262292431FS`:S`:Ole CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q@ c+m=122c+80=122c+80"80=122"80c=42 FMicrosoft Equation 3.0 DS Eq_1262293797FS`:S`:Ole CompObjfObjInfouation Equation.39qhh 4c+4p+6b=21.003c+2p+10b=20.882c+3p+4b=13.17 FMicrosoft Equation 3.0 DS EqEquation Native _1262293611FS`:S`:Ole CompObjfuation Equation.39q4h 4c+4p+6b=21.003c+2p+10b=20.88 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _1262293618 FS`:S`:Ole CompObjfObjInfoEquation Native )_1262293628FS`:S`: <9 ! FMicrosoft Equation 3.0 DS Equation Equation.39qpD 4c+4p+6b=21.00"2(3c+2pOle CompObjfObjInfoEquation Native +10b=20.88) FMicrosoft Equation 3.0 DS Equation Equation.39q <9 ! 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