System of Linear Equations- Practice Problems

8/3/2019 System of Linear Equations- Practice Problems

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Find the solution set of each systemof linear equationsby elimination. If the equationsare inconsistentor dependent,so state.Seeexample8 1 C.

356 Chapter Systemsf Linear qua't ions

2 x - y : 4x l Y : 51 0 . 6 x - 2 y : - 83 x - Y : 4

Example 2x - 3y : 23 x * 2 y : 3So lu t ion 2x - 3y :2 , r , , l , ,r . , r . r - -+

3x * 2Y : 3 " i i , , , . ,, , r - - )

Using2x - 3y : 2,2 - 3 y : 2 ; i r , i : 1 ' ; q , ; 1 1 i 12 - 3 y - 2 : , i r i , ' , ' i , r i.- 3 y : 0

v:0Thesolut ionet s f 1,0)) .

8 . 2 x I 3 y : 1 2x - y : 11 1 .x - 2 y : 6- 2 x i 4 y : - 1 2

4 x - 6 y : 49 x * 6 v : 9l 3 x

9 . 2 x - y : - 5x * 2 y : 01 2 . 2 x - y : - l3 x - Y : - l

3 8 . x - ( 0 . 5 ) y : 3( 0 . 9 ) x I y : - 0 2

: l 3 . , , , r 1x : I l l l i r L , r

r\\ / x - Y : rx - r y1 7 . 5 x - 3 y : l

) v - ' l , v : - S2 1 . 4 x - t y : 72 x 1 3 y : 6l \( ) 3 r - t y : 2'---z 9x * 3y : 62 9 .4 x - 2 y : 03 x - l 3 y : 5

1 ) q3 3 . : x * - v - - '2s ' , l 0

1 4 . 3 x - f y : t2 x - Y : 91 8 . * 5 x - y : 4- 5 x * 2 y 72 2 .4x - 3y 73 x - 2 Y : 62 6 . - x * 2 y : - 7- 2 x * 6 y : l3 0 . - 2 x - t y : 64 x - t Y : 1

\ 4 q3 4 . : x - v : '7 5 ' , l 0) ) ?

- l r : _7 " 5 ' l 0

( l s J x - r 4 y : 1 0x I 2 y : 43 x * 2 y : l lx - y : 2

2 3 . 8 x - y : - 44 x * 7 Y : - 3 22 7 . 6 x - 5 y : 0l 2 x - 1 0 y : - 3

l l. ; x * ; Y : Id z rl 2 l4 3 ' 1 2

/ ' t ( - l ?/ ' ^ - \ J L I(5 ) j Tx -y : )Y,, i x - t 1v : rJ J

1 6 .x - y : 33 x - Y : - 72 0 . 3 x * 4 y : 1 82 x - Y : [email protected] "y : lo/ 6 x - 2 y : 52 8 .x * y : 9- 4 x * 4 y : - 3 6

) 1l Z . , x - 4 y : 4Ix * 7 t :21 6 3- x - r2 5 ' 1 0

( 0 . 3 ) x - ( 0 . 8 ) y : 1 . 6(0 . r ) x*@.a)y : r .2 (0 . l ) x+ (0 .3 )y : -0 .3(O .s )x+ (0 .a )y :0 .7


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35 8 Chapter Systemsf Linear quations

Find the solution set of the following systems.Let p - . Substitute,solve or p and. , thensolve or x

6i . + a: ox ya*2:tx y

70. Jane nvested total of $4,000 n two accounts.72. Onenumber s 3 less han twice a second umber.74. The length of a rectangles 5 less han twice thewidth.

I76. A boat akes3i hourso go rompointA to pointBzand back.

2. (x * 2)(vz 2x * 4)4. Find hesolutionetof theradicalequation\/i\/TTa: 3.See ection-5.6. If three imes numbers ncreasedy 15, he esultis 51.What s henumber? ee ection-3.

l l- a n o s : ;andy.

l 165 . -+ - : 3xyl l : - )x y

69.7t .73.

75.

L_a:,x yL+L:,x y, l+ - : )v4

- 3x2x

68.

Using the variables andy, write an algebraicequation or eachverbal statement.Example Thesumof two numberss 36.Solution Let x : oneof the numbersand y : the othernumber., The algebraicequations x * y : 36.

The sumof twonumberss 502.Onenumber s 6 more han another umber.The length of a rectangles 4 more han three imesthe width.The differencen two electriccurrents s 33 amperes.

Review exercrsesIn problems-3, perform he ndicated perations. eesection -2.

l. (4x3 2xz+ l) - (5x3 3x2 x - 4)3. (5y - 2z)z5. Onenumbers 27more hananother umber. hesmallernumber s one-fourthof the larger number.

Find thenumbers. eesection -3.

8-2 t Applied problemsusing systemsof linear equations

3 Example 8-2 A

Beingable o solvea systemof linear equationsets us solvemanywordproblemsusing wo variablesas opposedo the one-variable pproachwe used n chapter2.l. Theperimeter f a rectangular lo t of ground s 238 meters. f the engthofthe rectangles l1 metersonger han wice he width,whatare hedimensionsf therectangle?Useperimeter 2(length)* 2(width).]

Let w : the width of the rectangleand Q the length of the rectangle.11 rneters longer han twice he widthength of the rectangle isa:

T h u s l - 2 w l l l ., = 2 w + 1 1l 1 + 2 ' w


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Section -2 AppliedProblems sing ystemsf Linear quations 359

We get the secondequationby using the formula P : 29 * 2w and the giveninformation thal P : 238.

1 ;. ; : 29 -l 2w ;i:riri;ri-rir: 'vv i :We form the system f linearequationsQ 2w -f ll2Q't 2w : 23 8Sinceoneequations already olvedor Q,we use he substitutionmethod.

'r l ;,r -:ll i ir '2,, . , , a ' i 2w : 238 i ' r .c; - rr:4w * 22 -t 2w : 23 8 lrirre'6w t 22: 2386w : 216w : 3 6NowQ:2w l l l: 2 (36 ) + 1 l: ' 12 + l l: 83

l , l i t l : t i ; r r t rt r t - i r t

The rectangle s 36 meterswide and 83 meters ong.Check : . 2 (36 ) + 2 (83 ) 72 + 166 : 238 i i . i

2 . (83) s 1 1more han 2(36) 1 r)Note We wi l l not show checks n the future.However, ou shouldalwayscheckyour answers.2. Arlene wishes o invest $5,000. If she nvestspart at 77osimple interest, part

at 67osimple nterest, and receivesa total interest of $332 after one year, howmuch doesshe nvestat eachrate?

Note We use he formula : pr t where is the s imple nterest eceivedwhenp r i nc i pa l i s nves t ed t ra t e lo r t yea rs . im ew i l l a lways e one yea r ntheseprohr lems,o we may use he simpl i f iedormula : pr '

Let x : the amount investedat '|Voand : the amount investedat 6Va. Neuse the following data table to set up equations.

Amountinvested Rate ofreturn AmountofreturnFirst nvestment x 7o/o 0.07xSecondnvestment v 60/o 0 06yTotal nvestment 5,000 l5 5 2 {The equationsre oundby reading ownthedata able seearrou's)- hesystem f linearequationss thenx + y : 5 , 0 0 00.07x * 0.06y-- 332


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36 0 Chapter Systemsf Linear quations

To eliminate decimal numbers n the secondequation, multiply each erm by100.Then, we have

-> -7x - 7y -- -35,0007x -f 6v : 33.200- y : - l ' 8 0 0y : 1 , 8 0 0

S incex - l y : 5 ,000 , t henx * : 5,000x : 3,200Arlene invests 1,800at 67o imple nterestan d $3,200at 770 imple nterest.

) Quick check a. The perimeter of a rectangular flower garden s 82 feet. If thelength of the rectangle s 5 feet longer than twice the width, what are thedimensions f the rectangle?b. Jim wishes o invest $7,500. If he investspart at 6Vo nterest, patt at 8Vointerest, and receives$536 total interes t after one year,how much did he invest ateach ate?

1 . Readhe problem areful lyndcompletely. otewhat nformat ionisgiven ndwhat nformat ion e wish o f ind.Wheneverossible,rawa diagram howinghe relat ionshipsn theoroblem.Choose i f ferent ar iablesor each nknown uant i ty.Useword statementsn he problemo writea system f l inearequations. enerally,herear easmanyequations s herear edifferent variables.Solvehe system f l inear quat ionssing neof the methods ouhave earned.

6. Checkhe resultsn he or iginal tatement f theproblem

Canyour Solve wordproblem sing ystems f l inear quat ionsn wovar iables?

x+y :5 ,0007x -t 6Y 33,200

2 .{4


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Sect ion-2 Appl ied roblemssing ystemsf L inear quat ions361

Set up a systemof two linear equationsand solve.Seeexample 8-2 A-1.Example Th e perimeterof a rectangular lowergarden s 82 feet. f the lengthof the rectangle s 5 feet ongerthan twice the width, what are the dimensionsof the rectangle?Solution Let w : the width of the rectangle and I : the length of the rectangle.

I : s + 2 'wUsingP : 2P+ 2w ,givenP 82, hen2 Q - t 2 w :Q : 2 w * 5Solving he system singsubstitution, ubstitrte2w * 5 for I in theequation Q 2w : 822 ( 2 w * 5 ) * 2 w : 8 24 w * 1 0 * 2 w : 8 26 w - F 1 0 : 8 2

6 w : 7 2 r ' r ' lw : 1 2 ; r i ' i i , : I , , r :

Q : 2 w l 5a: 2(.r2) sQ:24+s9 : 29The dimensionsf the rectangle re12 eetwide and29 feet ong.

If twice he engthof a rectangularloor sincreasedy three imes he width, he sum s 48feet.The perimeter f the room s 40 feet'Whatare he dimensionsf the floor?2. Thedistance rounda rectangularlowergardens64 feet. f the ength s three imes he width,whatare he dimensions?3. The perimeter f a rectangular lot of ground s 30meters. hree imes he engthminus our times hewidth s 3 meters. ind the engthand he widthoftheplot.4. A 20-footboardmustbe cut into two pieces o hatonepiece s 4 feet onger han heotherpiece.Howlong s eachpiece?

A 2l-foot pieceof pipe must be cut into tw o piecesso ha t on epiece s 9 feet onger ha n the otherpiece.How long is each pieceof pipe'l

6. The sum of two electriccurrents s 96 amps. f onecurrent s 22 amps es s han the other,how manyampsar e there n eachcurrent?

7. The sum of tw o voltages n an electriccircuit is 47voltsan d their difference s 25 volts. Find th evoltages.

8. The differencen the numberof teeth n two gearsis l4 and their su m is 72. How many teetharethere n eachgear?


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362 Chapter Systemsf L inear quat ions

Seeexample8-2 A 2.Example Jim wishes o invest$7,500. If he investspart at 670 nterest,part at 87o nterest, and rece ives$536 total

interest after oneyear,how much did he investat each rate?Solution Let x : the amount investedat 6Vo nterestand y : the amount investedat 87o nterest.

Amountinvested

Rateof return AmountofreturnFirst nvestment x 6Yo 0.06x

Secondnvestment 8Vo 0 08yTotal nvestment 7,500 536The system of linear equations s

x+y :7 ,5000 .06x *0 .08y -536Multiply he erms f thesecond

x * y : 7 , 5 0 06 x * 8 y : 5 3 , 6 0 0equation y 100.

-+ -8x * 8y : -60,0006 x * 8 y : 5 3 . 6 0 0- 6,4003,200-2x

U s i n g x * y : 7 , 5 0 0 ,* y : 7,500y : 4,300

Jim invested 3,200 I 6Vonlerest nd$4,300 t 89o nterest.Phoebewishes o invest$20,000, art at JVo nterestIand he estat 6 ^7o. l f her otal ncomerom heLtw o investmentsor on eyear s $1,370,ho w muchshouldshe nvestat each ate?

10. The income from two investments or one year is$1,485. f $19,000s nvested, art at 870 nd heIrest at 7;Vo,how much s invested t each ate?

all . The income ro m an 870 nvestments $300more

than the income from a 6% investment.How muchis invested t each ate if a total of $30,000 sinvested?

f 2.l Jamie nvests total of $16,000, ar t at 79o ntercsr- and oart at 97o nterest. f the income rom eachinvestments the same,how much moneydoesJamie nvestat each ate?

The income from two investments s the same. f$36,000 s invested,part at 7Toand the rest at 8Vo,how much is investedat each rate?Juanita investsa total of $22,000.She suffersa netyearly os sof $160on her two investments.f oneinvestmentmade her a 12Vo rofit and the otherinvestment aused n 8% oss.how much was neach nvestment?In one year, Simonemakes a l5Eo profit on oneinvestmentbut takesa l2%ooss on a secondinvestment. f she nvestsa total of $25,000andrealizes ne t gain of $1,320on her twoinvestments,how much is each nvestment?

13.

14.


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ExampleSolution

Section-2 Applied roblemssing ystemsf Linear quations 63

Sam'sPizzaEmporium ells wo kindsof largepizzasat $6.50and$8.00perpizza.If Sam sells 3 laryepizzasor $306.50, owmanyof eachkind of pizzadoes e sell?Let x : numberof $6.50pizzas old and : numberof $8.00pizzas old'Wenow set up a table o help us determine he equations.

Number fpazzas Costper pizza Amountofincome$6.50 izzas x 6. 0 6.50x$8.00 izzas v 8.00 8.00yTotalnumber 43 306.50The systemof linearequationss then

x - f Y : 4 36 . 50x * 8 . 001 r : 306 . 50Multiply thesecond quation y l0 to cleardecimalnumbers.

x - tY :4365x*80y :3 ,065

A keypunchoperatorat alocal firm works or $9perhourandan entry-levelypistworks or $6.50perhour.The totalpay or an 8-hourday s $476,and hereare wo more ypists hankeypunchoperators. ow manykeypunch perators oes hefirm employ?A clothingstoresellsmen'ssuitsat $152and$205persgit.Thestoresells32 suitsand akes n$5,659.How manysuitsat eachpricedoes he storesell?A hardwaresupplycompany ells wo typesofdoorknobs. he chromium-plated nob sellsat $8perknob,and he solidbrass nobsells or $11.50per knob.The company old420 doorknobsor$3,622.50. ow manyof each ypeweresold?A roadconstruction rewconsists f cat operatorsworkingat $90perdayand aborersworkingat $50perday.The total payrollperday s $1,600.fthereare 3 laborersdoingodd obs and 4 laborersare assignedo work with eachcat operator,howmanv aborers re here n the crew?

Mult ip lyy*6s + -65x - 65y : -2,79565x*80v : 3 .065l5 y : 270 Addmembersy : 1 8

Us ing theequat ion* y :43 ,x * y : 4 3

x * ( 18) : 43 Replacewi th 18x :25Sam old25pizzas t $6.50 nd18pizzas t $8.00'

16 . Skilledandunskilledwofkersareemployed y aconstructionirm. If 5 skilledworkers nd 8unskilledworkers re employed,he total wages erdayare$948.When3 skilledand 5 unskilledworkers re employed,he total wages er day are$580.What is the daily rate of payof each ype ofworker?The tickets or a puppet howcost$3.50 or adultsand$1.25 or children.f $853.75n tickets resold o anaudience f 503,howmanychildren'sticketsweresold?A movie heater old323 ickets or $831.50.fadult ickets ost$3 andchildren'sickets.cost$1.75, owmany icketsof each ypeweresold?Fernando assaved 3 coins n dimesandquarters.If he hassaved total of $7.15, owmanydimesand howmanyquarters oes e have?Pam has a collectionof fifty-two l3-cent and20-cent tamps.f the facevalueof hercollections$9.28, owmany2O-centtarnps oes hehave?

18 . 22.

23.19 .

24.


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364 Chapter Systemsf Linear quations

An automechanic as wo bottles f batteryacid.Onebottlecontains l07oacidsolution nd he othercontains 4Vo cidsolution.How manycubiccentimeterscm3)of eachsolution reneededo make120cubiccentimeters f a 6Vo cidsolution?,) LeI x : the numberof cubiccentimeters f l0Toacid solution nd-], the numberof cubiccentimeters f 4Voacid solution.

j: 1 r , .1 - , 1 - ;1 ,r:11- ' - i ,

Thesystemf inear quationssx+y :1200 .10x *0 .04y :72Toclear hedecimal umbersn thesecondquation, ultiply

r. l rtt: i tri-: ::,:,, :rl1ro,s urater. -hus, hei. - 1:l

' . . r : i i ' B I

each ermby 100.

A metallurgist wishes o form 2,000kilograms of 'an alloy that is 807ocopper.He is to obtai n thisalloy by fusing together somealloy that is 60Vopurecopper with some alloy that is 857opure copper.How many kilograms of each alloy must be used?How many grams of silver that is 60Vo ure mustbe mixed toge ther with silver that is 357opure toobtain a mixture of 90 srams of silver that is45Vopure?How many liters of a 3.5Vo olution and a 6Vosolution of acid must a chemist mix together toform 800 liters of a 4.57oacid solution?

24 0 Ad dmembers40 Div ide y6

28. How muchof eachsubstance ustbe mixedtogetherf a ewelerwisheso form 16ounces f65Vo ure goldfrom sourceshat are 50Vo nd 70Vopuregold?29. How muchof an lSVo altsolutionmustbe mixedwith 40 ml of a 30Vo alt solution o obtaina 25Vosalt solution?30. How muchpuresalt mustbe mixedwith 9 cubiccentimeters f 20Vo al t solutiono obtaina 4070salt solution?

x+y :120lOx 4y 720

Us ingx : - ' y : 120 ,x+Y :120

i1r{ r ) y : 120- / : 80

\ / i ) l i t i i \ , ' j t /

i : i i :p r l r r i, :wr l i r r,

-+ -4x - 4y : -480l O x * 4 v : 7 2 06x

Thus,40 cm3of lj%o acidsolutionmust be mixedwith 80 cm3of 4Vo cidsolutiono obtain 120cm3of6Voacidsolution.

26.

, 1

Volume o/oacid Amount ofacidFirst olut ion x 10 0.1 xSecondolut ion 4 0.04yTotalmixture 120 I 6 0.06(120) 72

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System of Linear Equations- Practice Problems