Key to Algebra Book 9 Systems of Equations

Key to

Ngeb*d Systemsof Eqaations

;*-

By fulie King and PeterRasmussen

^*.--*.j

TABLEOF CONTENTS S y s t e mosf E q u a t i o n. .s. . . . . . . . . Sofving Systems of Equations by Graphing ......... SolvingSystemsof Equations by Addition or Subtraction Solving Systems of Equations by Substitution.............. SolvingProblems S o l v i nS g y s t e mosf I n e q u a l i t i e s . . . . . . . . . W r i t t eW n ork P r a c t i cTee s t . . . . . . . .

...........1 .....................2 ......8 .................12 ...........28 ............30 . . . . . . . . . . . . .3. 5. . . . ..........36

Patternsof Primes The coverol this bookshowsthe lshangotallybone,whichis oneof the oldestmathematical artifactseverfound. Excavated in 1960inZaire,it datesbackto the periodwhenthefirstcities werebeginningto form,between9000e.c.and6500e.c. The lshangotally bone containsgroupsof notchesarrang€din threedistinctcolumns.Onecolumnof numbersis 11, 13,17, 19. Theseareprimenumbers.A numberis primeif its onlydivisorsareoneandthe numberitself.Hereis a listof the f i r s lt5 p r i m en u m b e r s2:, 3 , 5 , 7 , 1 ' l1, 3 ,1 7 ,1 9 ,2 3 ,2 9 , 3 13, 7 , 4' t,43,47,53. Primenumbershavefascinated humanityeversincetheir apparentdiscoveryin Africa.TheancientGreekswereableto answermanyquestionsaboutprimenumbers.Forinstance, they provedthatthereare infinitelymanyol them. The searchfor orderand pattemis oneol humanity's basicdrives. lt is alsooneof the basicelementsof mathemat ics. However,althoughthe searchfor a patternin theappearance of primenumbersis intriguing,the answerhas provedto be elusive. Lookat the list of the first 15 primenumbers.Canyoutellwhatthe 'l6th numberis withouttestingits divisibility?The Greekssawno way; neitherhavemathematicians in the 2000yearssincethen. Humanimagination andcreativity haveprovidedsomecluesabout 2 3 4 5 6 the natureof primenumbers,as this 7 I I 10 tl 12 speciallistingof the numbersfrom2 t 3 1 4 1 5 1 6 1 7 1 8 to 102shows.Theprimesareprinted t 9 20 21 22 23 24 in bold. What conclusiondo you draw? 25 26 27 28 4' 30 Whatissoimportantaboutprime 3l 32 33 34 35 36 numbers? 37 38 39 40 4 l 42 Foronethingtheyformthebuild43 M 45 46 47 48 ing blocksfor all numbersin exactly the samewaythatthe chemicalele49 50 5 1 52 53 54 mentsformthe buildingblocksfor all 55 56 57 58 59 60 matter.Forinstance, thenumber126 6l 62 63 64 65 66 i6l9! og o$ o/. canbe broken6ls1v6 67 68 69 70 7 1 72 It cannotbe brokendownanyfurther, 73 74 75 76 77 78 and this is the only way to breakit downintoprimes. 79 80 8'l 82 83 84 Fora longtimeth€ importiance 85 86 87 88 89 90 of primenumberswasmostlylimited 9 1 92 93 94 95 96 to heir roleas buildingblocks. But inthe1980'swhen cryp97 98 99 100 t o l 102 thatchanged foundnewusesforthem. tographers

Historicalnoteby DavidZtarelli lllustration by Jay Florn arepeoplewhosendsecretcod€s.Atthebeginning Cryptographers of eachcodethereis a'kef that enableslhe receiverof the codeto decipherit. Thekeysare largenumbersthatcan be tacloredintoprime numbers.For instance,a cryptographer can senda codewiththe key 267-'1. ll the personwho readsthe cod€knowsthat 267- 1 = 193,707 ,721t 761,838,257 ,287 thenthatpersoncandecodeit. who writessocretcodesthereis another Foreverycryptographer whotriesto brsakthem. Wh€nbankswiretheirfinancial cryptographer at the closeof the day,theywantto be surethatthe codes transactions are secure. This meansmakingsurethat the k€y is not known. For mathematicians, this meanslhe searchfor biggorprirn€numbersand moreeflicientwaysof factoringnumbersintoprirnes. Youcan restassuredthatyourmoneyis safe- prirnefindersare aheadof numberfactorers.The governmenfss€nsitivemessagesare ona bone guardedtoo. Allwithnumbersthatmadetheirfirstapp€arance yearsago. in Africaabout'10,000

IMPORTANTNOTICE:This book is sold as a studentworkbookand is notto be used as a duplicating master. No part of this book may be reproducedin any form without the prior written permissionof the publisher. Copyrightinfringementis a violationof FederalLaw. Copyright@1992by KeyCurriculumProject,Inc.All rightsreserved. @Key to Fractions,Key to Decimals,Key to PercenE,Key to Algebra,Key to Geometry,Key to Measurement,and Press. Key to Mefic Measurement arc registered trademarks of KeyCurriculum Published by KeyCurriculum Press,115065thStreet,Emeryville, CA 94608 lsBN 1-55953-00$X 21 20 19 08 07 06 05 Printedin the UnitedStatesof America

Systems of Equations Youhavejusttunedin to thefootballgame.Yourteamis winning27-18.Theannouncer saystheyhavescoredfivetimesonfieldgoalsandtouchdowns andhaven'tmisseda point-after-touchdown yet. Ganyoutell howmanytouchdowns andfieldgoalsyourteam hasmade? Infootball a touchdown andextrapointareworth7 pointsanda fieldgoalis worth3 points. Youcanprobably figureouthowmanyof eachtheyhaveby guessing or makinga listof possibilities. Tryit here.

Whatif youtriedto usealgebra instead r andf of guessing? Youcouldbeginby choosing as variables.

touchdowns Nutnbut

Points

t 7t

fi.ld Stlg

totaI

+

5

3f

27

Thenyou couldwrite They scored5 tirncs.

t + +=5 o

" w 7I * 3f = 27o" ffi

To findthe numberof touchdowns andfieldgoalsall you wouldhaveto do is to finda single pairof numberswhicharesolutionsof bothequations.Canyou do it? In this bookwe will lookat severaldifferentwaysto solvesystems of equationslikethis one,andat waysto solvesystemscontaining inequalities, too. o19@by KsyCuniculum Press,Inc. Do nol duplicatswithoutpormission.

1

SolvingSystemsof Equationsby Graphing Remember thateachsolutionof an equationlike t + f = 5 corresponds to a pointon its graph.Thesameis truefor solutions of 7t + 31=27. lt we couldfinda pointwhichis on bothgraphs,we wouldhavea pairof numberswhichis a solutionto bothequations. So . . . let'sgraphthe equations! Thefirstequationis easyto graphby plottingpoints.

Wewillsolvethesecondequationtor f to finditsslopeandfintercept.

7t-'\ 3f ='*27

f=5

t t

t

+

U=-!L*fr I 3 { = - 3 ++ g

+

slope:j

f-inter"ept'?

--- +- -- 1--- --t--- --{--

-- t-- -.''1- ----+ ----{--.

--l-----t----+---t-.

! ^ L

|

|

!

..:.tlit i

i _l

" 'T

"i'-'i

i

i

i

i

:

' ' "'i""'i""'t"-'i"" ------+-----t-----t-.---+--. l

l

:

:

"""r""'i""'i"' ri . i i i

I i-i I i i I i----.-+......i....-.i.....+--. i

i

i

i

...---+-----:------l-----+----

: : : : ""--1""'1"""t""'*-" i

i

i

i

Thegraphscrossat the point(3,2),so (3,2)is a solutionof bothequations!We cancheck thisby substituting 3 and 2tor t andf in thetwoequations.

t + f = 5

3*2=5

7++3f=27 7 . 3 * 3 . 2 = 2+1$ = 2 7

Didyoufigureouton page1 thattheteammusthavemadethreetouchdowns andtwofield goals? 01992 by Key Curiculum Press, Inc. Do not duplicats withoul p€rmission.

Drawa graphof eachequation.Choosewhatevermethodof graphingyou findeasiest. Writethe equation alongthe lineandlabelthe pointwherethe graphsintersect.Then checkto makesurethe coordinates of thispointsolvebothequations.

?( 'l =7 3 x + y = l

I Check:

x +

Y

=6

3y x = 6

lllliii--liicj

-.---i----i----i-.--i--i--.-i---.t--..i-...:....ri

i

:

i

:

i

l

:

i

J

.-?$ i ...i...j:.s...i....i....i....i... -----i---i---;---i---i-j-.-;---i-..i-...i--

Check:

@19Cby KeyCurriculum Press,lnc Do notduplicata withoutpsrmission

" " ' ii' - - -i i" ' -ii " " ii" " :i " " ti" "i i " " ii" " :i- "-' ?ae B

3

Solveeachsystemof equations by graphing.Checkthesolutionbysubstituting.

2x + y = 3

3y=^ 12

Y

Y

=-x + Ll

= € " *2

a y - 3 x= 1 2 Y

+ 2x=-B

v + 3x =-2

2y - 3 x = l +

olg by Kry Curlculrm PBss, Inc. Do nol d$licde wlthoutpennlssbn.

Someof theseequations arenotlinear,butyoucanstillfindthesolutions foreachsystem by graphing.Besureto choosebothpositive numberslorx. andnegative

y=x' Y

= - 2 x+ l

y =l ^ l y =Ix +J

-. -i----t -- --" --- -i----'... -i -..i. -. t-.

'_i____i____i____t____i

-i'___i-_--i_-__i___.t-..-i-.--i----i'i---!--)---t---i---i---t---t-

Thesolutionsare o o

o o

x =y"-q t

@19eby KeyCurriculum Pross,Inc. Do notduphcate withoutpormissbn.

=2,1 |

Herearesomemorefootballscores.Writetwo equationsyou coulduseto findthe number of touchdowns andfieldgoalseachteamhasif all extrapointsaftertouchdowns were made.Usegraphsto helpyou if you needthem. COUGARS30 (Scored6 times) Equations:

touchdowns

fieldgoals

OWLS24 (Scored4 times) Equations:

touchdowns_

6

fieldgoals

01992 by Key Curiculum Press, Inc. Do nol duplicals withoul p€rmission.

Writea pairof equations foreachnumberpuzzleandfindthesolution.User forthefirst number andy forthesecondnumber.Drawgraphsif youthinktheywillhelpyou. Guessmynumbers. Thefirstnumber istwomorethanthesecondnumber.WhenI add twotimesthefirstnumber to fivetimesthesecondI get25. Equations: ---i--.i-.--l----:----t...-1.-..1...-i----i----,--. "-'i-"i---i---t---i--t--t-"t"-r--:-' ..- - -4..--,.----,,----"---.!--.-1.--:-..r-.--l----,--

'----t----|----!----i----i----!----i----t----i---.---

"-l----i""1_"_i'_..i-_.1.'.-t.-.-t----i-------.-

"-"i.-.r-.-i----i----i----i----i..--i----i.-.-.---

-.---i----)---l---i---t----!----)----l----t------

The numbersare

and

I'mthinkingof two numbers.Thefirstnumberis threetimesthe second.Thesecond numberis fivelessthantwo timesthe first. what are my numbers?

Equations: --:----l----t----t----l----1----j----i----t----i-----

'-i'--i""i-"-i--"i"-'i-"-r'-r-'i-"-r-"...i.-...:.._.:-.._i-_..:.-...i_...:....i....,i_...i...._

Thenumbers are @199by KeyCurriculum Pross,Inc. Do nol duplicate wilhoutpsrmission.

and

7

SolvingSystemsof Equationsby Additionor Subtraction 65-43.TheAceshavescored36 timesand TheAcesare beatingthe Rocketsin basketball haveno three-point baskets.Howmanytwo-pointbasketsand how manyfreethrows (onepoint)havetheymade? To sotvethisproblemwe couldchooseb torthe numberof two-pointbasketsandf for the numberof freethrowsmade. Thenwe couldwritethe equations.

b+f =36"'" ?b" f =65"oo wayto solvethissystem.Doyouseewhy? wouldnotbe a practical Graphing easywayto findit. Wecanjust butthere'sanoth6r Wecouldtryto guessthesotution, sideof theother. fromthematching subtract eachsideof oneequation 2b + f = 65 [+ { = 36

f,+O= 2q 29forb in is29. Wecansubstitute hasonlythevariable b. ltssolution Ournewequation to findf. oneof the two originalequations

f,+ f = 36 ? ? t f = 3 6

f = 7 TheAceshavescoredon 29two-point (29,7)isthesolution to oursystemof equations. foryouto try: basketsand7freethrows.Hereis a similarproblem baskets andfreethrowsto earn39 points. TheExplorers made21two-point youcoulduseto tindouthowmanyof eachtheyhavemade. Writetwoequations method. Trysolvingyourequations bythesubtraction Equations:

baskets two-point I

freethrows Prass,Inc. o1S2 by KoyCurriculum withoutpemission. Do notduplicate

Thesolutions wouldbe hardto guess,butyoucansolve to thesesystems of equations method. eachsystembythesubtraction

3b+f =2? [ + f = -1 7 -

2 b +Q = l ?

2b= 12

i " b =b 6 + f = t7 f = ll

Wc ccn chccl our

5 x * y= 2 3

solrtion h rnbstit,^ling 6forbaol lltrf

Zx+y=lq

in lhc othcr rgectim,

o o

o" chcch:

3'6+ ll =2j f 8+ l l = 2 j

21=21

Thc solutionis ( 6 ,l l 1 .

7 ^ * 2 y =l 8 3 x * 2 , 1= 2

5x-3y=32 2x - 3y= ll

Ty =-6 ? ( + y = 3

6x

X +

@199by KsyCuriculumPrsss,Inc. Donotduplicats wilhoutp€rmission.

?(

5y = - 4 5y = 1 6

I

Eachof thesesystemsof equationscan be solvedby additioninsteadof subtraction.

Z x * y= 7,"offi-*Q

3, - y =-irz

5r * O=-5 5r =-5 r =-l 2(l)*\ =7 _ 2 * \= J y=l

oppofteJ (T". vt-,r--l:r.-:

x Y = 1 2 5x* y = 0

Check:

3 ( - l )- 9 = - | ' 2 -3 - I =-12 -12=-12

The sol..tionis (-1,9).

4r - 3y =-26 x*3y= I

-2x + 3y = 1 7 ? 2x+

3 x - +y= 2 5 3 x +ay = - 7

- x- 3 y = l l 3x * 3y =-21

10

v

@1902 by KsyCurriculum Prsss,Inc. Do notduplicalewithoutpermission.

Solveeachsystemby additionor subtraction. .1 oo%oo

x + L y = Z ( The y-terns are oppositcs. f con

4 x - 2 y = 3 8 rdd

to get O.

o

Jx

o ot / o o

y =lo I

2x-y=12

7 x * y = -2 7x*3y= B

4 x - 3 y =8 4x *3y= 32

x .+ 7 y = 2 1 ^ - 2y = -6

"tr+ 6y - -f x 6y =-13

Press,Inc. @19Cby KsyCurriculum Do notduplbatewithoutp€rmission.

The y-terns orc thc sanc. If I sb{rcct I1l g:t O.

11

Solving Systems of Equations by Substitution Thissystemisn'tsetupto solveby additionor subtraction.

= Y 4x-l 3x+2y=1 Thereis a wayto solveit whichis easierthaneitherthegraphing orthe addition/subtraction method.lt is substitution. To solvea systemof equations withan we replaceoneof the variables by substitution equivalent expression containing onlytheothervariable.

y = 3r+20t ?

ooo

Tlrir qrrtbr

{r - |

tc I cra rcplrcr

3x+ 2(4x- l)= I

3x* 8x llr

rqt

y uth {r-lir

2 =I 2 l l x= l l x = l

Nowwe havea solutionforonevariable. Wecansubstitute thisnumberin oneof theoriginal equations to geta solution for theothervariable.

Wecheckoursolution our by substituting pairof numbers in theotheroriginal equation. Check:

3x+2y=q 3 ' l * 2 ' 3 =q 3 + 6 = 9

}/={x-l

y = + ' l- | y =3

Thesolution is ( 1,3). 9 = ? Circlean expression andthevariableyoucouldsubstitute it for in eachsystembelow.

Y=@ ^ *OTH 2 x - 3 y= 9 x =2y + 2 12

x = y-3

/

= -tlx

5r*3I= l

6x*y = 6

Y = 2 ?( 5x+4/=5

v= 5 x - 1 x

= Y+

5

ol$? by KEyCuniculum Pro3s,Inc. Oo not duplinte withoutpermi6sion.

Nowgo aheadandsolveeachsystembysubstitution.

Y=E

x * f l =l 4

t + 3l2xl= I t l , + 6 x = lrt 7 x - Irt t : 2 l=2r

x = rl Check: r+3y=lt 2 + 3f0= ltl / + 1 2= l t lrt = lrt

3

5 x* 3 y = l

'f =zel l=+

Qr+'

b x * y= 6

2 x - 3 , 1= ? x = 2'l *2

y= 2 - x

'l = 5 x - 1

5x*tfl =

x = I + 5

Y

=-4x

6'199 by KeyCurriculum Press,Inc, Oo not dupli)atewithoutpennb.ion.

13

Solveeachsystemof equations bysubstitution.

y=@

2 x- f f =z

2x-(3+rl=2 2r-3 - * =I x-3 =/ r = 5 l= !+ 5

g Y=

Check:

3x-l=7 = 2x - t+ I

2r-Y=2 2.5-8=2 l0-6=2 2=2

Thesolutionis ( 5, E ).

^ = 2 1+ 7 3 , -1^ = 8

y=lO-2x

3x-2y=22

14

Y = x - 1 2x-3y=ll

,l=5Ox+t-[ y - 32x = LiO

by K€yCuniculum Pross,Inc. @1992 Do notduplicalawithoutp€rmission.

you firstwill haveto solveone equationfor To solveeachof thesesystemsby substitution

oneof thevariables.

x + 'l = l2oo'l

2x + 5 y =2 7

^-3y=12 2x * 5y =-2O

i

2@-+5y = 27

t l y =o 3x* 2 y = 2 0 x *

@199 by Koy CurriculumPrsss,Inc Do not duplicale without permissbn.

y x= 1 5 2x * 5,1= 2 6

15

Solveeachsystemon thesetwo pagesby the methodyouthinkwillbe easiest:graphing, addition, subtraction or substitution. Useoneof the gridson page17 if youchoose "See graphing.Write graph"underthe problem.Writeeachequationalongitsgraph. Remember to checkyoursolutions.

y = a+

$

2 ^ - Y= - 1

y=3x-l y = 2x +2

y=fx*6 3x+y=-7

16

x + ,f =-t+

-x *2Y = 13

x + y = 20 30r+25/ = 550

- 2 y =6 2x + Y =7 x

Prsss,Inc @1992 by KeyCurriculum Do notduplbatewithoutpermission.

x +3/=|

2x " 3I =-t+

Y=2x-7 x*2f=ll

7 x+ 5 y= 2

2x * 5I = |

14 y=2

2x y=9

olg by KeyCurrlculrmPr€3!, Inc, Do not dupllcalewlthoutpermltclor..

17

foryouto solve.Foreachproblemchoosetwovariables Herearesomemoreproblems foreachproblem.Usethe andsaywhattheystandfor. Thenwritea systemof equations solution to answerthequestion. 'TheHawksstillhavea narrowedgeoverthe Pioneers game.They're in thisbasketball only29. Sofarneither havescored30times;theHawks, leading 51to 49. ThePioneers baskets.' teamhasmadeanythree-point Howmanytwo-pointbasketsand how Howmanytwo-point basketsandhow manyfreethrowshavethe Hawksmade? manyfreethrowshavethe Pioneersmade? Vaiables: Variables, ! := nlnrbcrof bocltctr -.:-f : Equaflon:

nurnb.: cf frce throyr

Equation:

Answer:

Answer:

Nickysaid,You'llneverguessmy numbers! Theirsumis 12andtheir is 26.' Aretheretwosuch ditference numbers?lf so,whatarethey? Variables: Equation:

Tonisaid,"l betyoucan'tguessmy numbers!Thefirstis fivemorethanthe second.lf youaddthefirstto threetimes thesecondyouget29'. WhatareToni's numbers?

Answer:

Answer:

18

Olg by K€f CurlcubmPres8,Inc. Do nd dryllcale wtlun pemlssbn.

A 96-inchboardhasto be cutintotwo pieces,onepieceI incheslongerthan theother.Howlongshouldeachpiece be? Variables: Equation:

ropeinto Howcouldyoucuta SO-meter threepieces-twoof thesamelengthand thethird,8 meterslongerthantheother two? Vartabbs:

Terrydidall25 problems onthetestand gotan 85. Scoresarecomputed by giving4 pointsfor eachrightanswerand 1 pointforeachwrong subtracting didTerry answer.Howmanyproblems getrightandwrong? Variables: Equation:

Linpaid426tora pencilandtwoerasers. andthree Kimpaid910forfivepencils erasers?Whatwerethe pricesof pencils anderasers? Vaiables:

O19e bt Ket CurdcubmPrss, Inc. Do nd d$lbaie wtthoutpefldsslon.

19

Wealreadyknowhowto solvemanysystemsof linearequations.Thesystembelow cannotbe solvedquicklyby anyof the methods we havelearned,butwe canchangeit into a systemthat'seasyto solve.Allwe haveto do is usethe Multiplication Principle to multiply bothsidesof thefirstequation by 2.

5x + 4y = l,{

3x-lV=26

'2

5x + 4,1= l+

,6x-4y=SZ

5'5+4y=lL+ 30+4y=lt+

4Y=-16

llx+ 0 =66

oo

Y= - +

llr = 66

x = 6 fhe solutionis (6,-4).

Replaceeachsystemof equationsbelowwithan equivalentsystemwhichyou couldsolve by additionor subtraction.Youdon'tneedto solvethe system.

7 x+ 6 y = 2 ^ 7 x * 6 y = 2 2^^-3J = lO 'z >4x - 6; = 20 /r.ataraa,-.r-

@

3x+ 4y = ll

3x+5y=26

2x- y =13 =-12 ^ - 2v I

5 x * 2 y= - 5

3x*8y=3+

4x*3y=7 2x-9y=35

2x 3 I = o -28 6 x 5y =

7x-3y=37 2x- y =12

5x-4y=lO

x +

|

=

lo

l5r * 2 g y = l|75

3 x - 2 y= 6

*2Y=21 9x * 2+y= 243 x

r,+ y =20 3h *'t4y= Bl2 20

=1260 60x*75,1 o1g by l(et CunlculrmPrBs, Inc" Oo noi duplbalewlthod perrl€.lon.

Solveby additionor subtraction.

7x+6y=2 2x - 3y = lO

3x + 5y=26 2x y =13

x

2y =-12

3x+ay=lf 5x * 2y =-5

3r+8),=3Ll

4x"3y=7 2x-ql=35

2x*3y 0 6x - 5y =-28

Press,Inc. @199by KeyCutriculum Do notdupli:atswilhoutp€rmisshn.

21

Solveeachsystemof equations.A calculator maybe helpful.

7 x - 3 y= 3 7

5x try t0

a

3x- 2y

Lx-

Y

= rl ta-

x + , 1= 1 0

l5r * 28y= 1176

'l = 2 x 3lx * 44y= Bl2

22

b

,r,,+ 2y = 2l

1r+Z+y=243

? ( + l = 2 0

60x*75y=1260

@1S2by K€yCurriculum Press,Inc. Do notduplicate withoutp€rmission.

Then Multiply oneequation by 10or 100so thatyoucansolvebyaddingor subtracting. pointone multiplying by 10movesthedecimal solvethesystem.(Remember, a decimal spaceto theright.Multiplying by 100movesit twospacesto theright.)

tlx+y=50 T x + I = 5 0 .3x+.2y=2.5 rlo, 3f, - I = 3tl .3x-.ly = 3.+ x 2 Y= - 5

.7x + . 5 y = B

2 x - 3y = 4 . 0 2 x+. O l y=. 5 2

. 0 5 x* . 1 0 y= 3 5 x +5 y = t 6 5

. 2 5 x + .05y= 2.75 5 x +5y=95

X +

5 y= 2 0

@1992 by KeyCurriculum Pross,Inc. Do nol duplbalswithoutpermission.

23

Sometimes we haveto multiplybothequations by differentnumbers to geta systemwhich canbesolvedby addingor subtracting.

Tx+3y=12

5 x * 2 y= |

x2

x3

t+(-3 * 3) y = 1 2 -12* 3y = 12 3'l = 2+ y =I

Bx + 6y = 2+ f5x + 6v = 3 -7x + 0 = 2l -7r = 2 l

-3

x =

The solution is (-3,8).

Writea pairof equivalent equations whichcanbesolvedby addingor subtracting.

2x+3y=10 3x - 4y =-2

? x - 2 y= 1 5 4 x + 3 y= - 5

7x-6y=lO 2x*5y=23

2 x - l l y= 1 5 5n*3y=7

6x - 5y = l 6 4 r - 3 y = 12

?x - lOy= 7 5r+8I=31

2x-3y=50 7x + 8y=-10

5x*6y=16 3x- 4y= 2

3 x + 2 1 t+ = 2 x + 3 y =I

2 x + 3y = 1 0

5r

lOx*3y=15 3x - 2y =-lO

5 x * 2 y= q 2x* 3y=-3

24

o o o O

ay=

2

Otg by K.y Cunlculum Pr63. Inc, tlo rd dljpllc.L wllhoulp.fi{..lon.

Solveeachsystemby addingor subtracting

2x*3y=10 3 x - 4 y= - 2

7x

6Y= lo

2 x + 5 y= 2 3

6x- 5 y = 1 6 tlx - 3 y = 1 2

O19@by KeyCurriculum P€ss, Inc. Do notduplicate wilhoutpermissbn.

?r,- 2,1= 15 4 x + 3 ) ,= - 5

2 x - l l y= 1 5 5 x + 3 y= 7

1x-l O y = 7 5 x +8 y = 3 1

25

Solveby addingor subtracting.

2x - 3y = 50 7x+8y=-10

5 x +6 , = 1tb 3 x - 4 y =2

3x*2y= 14 2 x* 3 y = I

2x*3y=10 5 x - 4 y= 2

lOx*3),=15 3x - 2y =-10

5 x + 2 ,=11 2 x+ 3 y= - 3

26

@1992by Koy CurrbulumPress,Inc 0o not duplicale withoul permissbn.

andby substitution. Solveeachsystemtwoways:by addingor subtracting

x

7Y = t+

3x + y =-10

4x*y=3

5y -t1

Whichmethodwas easier? methods? yougot by different Whatdidyou noticeaboutthe solutions

@19e by Ksy CurnculumPrsss,Inc. Do not duDlicalswithout psrmission

27

Solving Problems Choose twovariables andwritea pairof equations to solveeachproblem.Thenuseyour solution to answer thequestion problem. inthe 85 vehicleswitha totalof 188wheels registered for the bike/trike race. Howmanywerebicyclesand howmany weretricycles? Variables: Equations:

A largebowtakes5 feetof ribbonanda smallbowtakes3 feet. 150feetof ribbonis available to make36 bows. Howmanybows canbe largeandhowmanymustbe small? Vaiables:

Answer: Tapesare on salefor $8 each. CD'sare on salefor $9 each. Renespent$110of herbirthdaymoneyandbought13 sale items. Howmanytapesand howmany CD'sdidshebuy? Variables: Equations:

A collection of 31 nickelsanddimeshas a valueof $2.65. Howmanynickelsand howmanydimesarethere?

Answer:

28

@1992by Ksy CutriculumPress,Inc. Do nol duplicate wilhout psrmission.

othermethodseems for youto solve.Usealgebraor whatever Herearesomepuzzles mostefficient to you. "Myson'syearsandmine andsome A farmerhadsomechickens 40 headsand126 cows.Shecounted andhowmany legs.Howmanychickens cowsdid shehave?

Makefifty-nine. Whenhe cameto be. I wasjusttwenty-three. Howold arewe?"

Answer:

Answer:

Jamiand Mamitogetherhad$32. Jami said,"lf you giveme $5, we'llhavethe sameamount."Howmuchdadeach one have?

A birdflew 12 milesin an houraidedby a tailwind.Thenit turnedandflewback againstthe windbut was ableto fly only 4 milesper hour. Whatwas the speedof the wind? Howfast wouldthe birdhave flownif therehad beenno wind?

Answer:

Answer:

How manyof the puzzlesdid you solveby writinga systemof equations? O19SAby l(ey Cl,riolum Pr3ss,Inc. Do nd d.dlcab rithout pemi$ion.

29

SolvingSystemsof lnequalities To solvea systemof inequalities we cangraphbothinequalities using thesamepairof axes. Lookat the graphto the rightto see howwe wouldsolvethissystem:

V < I V > T Oneregionhasbeenshadedtwice. Thepoint(4,2)is in thisregion.lts coordinates makebothinequalities trueas shownin thetablebelow,so (4,2)is a solution.Nowyoutestthe otherpointsin the tablebelowto see if theyaresolutions.

( x , y)

(4,2)

rs (r,y) in thc dcuble- rhodcd rcgion?

fs (r,y)

3 -2x

z < ++L 2< 6

yes

o rolrtion ?

2 > ? -2(.+) yes 2 >5

Yes

(6,-l) ( 2,-5)

(-5,3) ( 3 , 5) hck o point.

Didyou noticethatall pointsin the double-shaded regionaresolutions of bothinequalities andpointsin otherregionsare not. Thedouble-shaded regionis the graphof the solution setforthissystem.Sincewe couldnotpossibly listallthe pointsin thisregion,the graphis the easiestwayto showthe solutionset.

30

@1992by Kay CuiliculumPrsss,Inc. Do not duplicats without permissbn.

setforeachsystemof inequalities. Showthesolution

v

Y

v) - Z x +2

v

Y <

,l

Y >

v

o19e by KeyCufrlcubmPr68, Inc, Do nd dupll)atewithoutpermissbn.

31

Graphthesolution set. o

o

x >

o

The grcphof r = 3 is o vcrticol line. Pointsto the right hove r-valucs gce&et thcn 3.

Y < --'t---t--:--'t--t--t--:--'l

x

>

-l x t

32

v v

Pr6B, Inc. olSA by KeyCuniculum Do not dudlcatewilhoutp€rmlssion.

Herearesomewiththree. canhavemorethantwoinequalities. Systems of inequalities To findthesolution setforthesystemlookfortheregionwhichis shadedthreetimes.

x Y

v

y>2x-10 y >

Y> - l x - 6

olg bt Kst Currlc1Jlum Pr6s, Inc, Do rd (ldbde wnhoutpernission.

v v

y < - 3 x +4

y < $ " +l0 y < - ? ^ + Lt

33

Twoof thesolution setsonpage33weretriangles. Thesolutions setsonthispageinclude a square, a rectangle, a parallelogram anda pentagon. whichsolution will Canyoupredict bethepentagon? y> *x y>-2x-lO

Y < -2x +6 y s

'-r----r--.-i.---i"---i----i---'i--.

y > x - 3 ' 1. t + 6

r-5 x + 5

y )-x-5 ,l < -x + 5

ffi

34

@1992by Key Cuniculum Press, Inc. Do not duplicats without pormlssion.

WrittenWork Dotheseproblemson somecleanpaper.Labeleachpageof yourworkwith yourname,yourclass,the date,andthe booknuniber.Alsonumbereachproblem. Keepthiswrittenworkinsideyourbook,andturnit in withyourbookwhenyou arefinished. Pleasedo a neatjob. 1. (5,2)is a solutionto oneof thesesystemsof equations.Whichone? 3r+y=13 x+3y=-2

?-x+3y=! x-Y = 7

x-2y='l 2x-y=g

x = y2-1 J = r+3

2. Solvethissystemof equationsin threedifferentways. 2x-Y=7 3 x + 2 Y= 1 4

hasno solution: 3. Thissystemof equations Y=2x+5 2x-Y=4

to seewhy. a) Graphthe equations by substitution? b) Whathappensif youtryto solvethe equations hasmanysolutions: 4. Thissystemof equations x+4Y=6 3x+12Y=l$

a) Whathappensif yougraphthe equations? b) Whathappensif youtry to solveit by subtraction? c) Namethreesolutions. 5. Thesolutionsetof a systemof inequalities is shownto the right. a) Namethreesolutions. b) Namethreeorderedpairswhichare notsolutions. Why? c) ls (3,2)a solution? 6. Solveeachproblemusinga systemof equations. a) Howcouldyou make980usingexactly 30 nickelsandpennies? b) Fivepencilsandfoureraserscost980. Two pencilsandthreeeraserscost490. Howmuchis a pencil? olgP Dt KoyCurriculum Prsss,Inc. Do rd dtplbalowithoulpermiss'ron.

3 5

PracticeTest Solveby graphing.

Solveby addition.

2 y - x = 8

3^- y =-f3 Z x +y = 3

Y-2x=-/ +---i---+- --i.-- +- --t -. +..-i ---+- --i ---+- --i.. . + - - - t - - - j - - - - i - - -

+---l---+---t---+-.-l- - +- '-l--- +- -+ --- -

+--{---+---l-.{..--t-. .l---{---1---+---. t- --t --- l-- -.1--. +. --t - ' r - - - l - - l - - . { ' - - --t---+- --r---a - -1.-. +- --i-- +--l--- -r-.

+- --i --- t- --t --. , , - -- . I - - - i - - - - | - - - l - - - - I

' - l - - - - f- - - i - - - + - - . i - - - - t - . - t . - - - l - - - t - . ' - t - . t-. -t - -- 1----t --. J . - - . - t- - - t -

--t --- 1----l

- . i . - - - t - ' ' 1 - - - + - . . 1 - - - + - - - t - - - - l - - - t - - - - l - .+ - - - l - - 1 - - - . t - -J.. - - - r - - - i - - - - l - - - i - - - { -. r----l-.-l ----f- --t---+---t --. t- -.t --'+-.

l---t--- t. --t --. t -

--t --- r'--I

- - - 1 - -- - l

- - t - . - - r . - - i . - - - -f - t - - - + - - - t - ' - t - - - t - - - - t - .

Solveby additionor subtraction.

*x * 3y=ll 2x + y = 7

2x + 5y = 13

3 x- 4 y = 8

Solveby substitution.

x = 2 y- 3 4x+3y=32

36

3 x * y =lo x - 2 Y= I

@'lS2 by KsyCurriculum Pross,Inc. withoutpormission. Do notduplicate

Showthe solutionset of eachsystemof inequalities.

y > y 3 - 2 x +2

Answerthe questionin eachproblemby writinga systemof equationsandsolvingit.

A football teamhasscored34 pointson

parkcost I nOunticketsto theamusement

ticketscost$5. lf 14 kids andfieldgoals.Theyhave I Sgandchildren's 6 touchdowns wereadmittedfor $82,how manywere madeeverypointaftertouchdown.

theadultpriceandhowmanygot havetheymade? | charged Howmanytouchdowns in at the child'sprice?

o199 by KeyCutftulumPrcss,Inc. 0o not duplicatewithoutpermis€bn.

37

Book l: Operationson Integers Book 2: Vsriables,Termsand Expressions Book 3: Equations Book 4: Polynomiq.Is Book 5: Rstionsl Numbers Book 6: Multiplying and Dividing Rational Expressions Book 7: Addingand Subtracting Rational Expressions Book 8: Graphs Book 9: Systemsof Equations Book lO: SquareRaotsand QuadraticEquations Answersand Notesfor Books 1-4 Answersand Notesfor Books5-7 Answersand Notesfor Books8-1O

Key to Key to Key to Key to Key to Key to

Fractions@ Decimals@ Percents@ Geometry@ Measurement@ Metric Measurement@

ilh

PRESS CURRICULUM KEY v Innovatorsin MathematicsEducationrsBN 1-55953-009-X

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