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,
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lsBN 1-55953-00$X
21 20 19
08 07 06 05
Printedin the UnitedStatesof America
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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
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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.
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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:
" " ' ii' - - -i i" ' -ii " " ii" " :i " " ti" "i i " " ii" " :i- "-' ?ae B
3
@19Cby KeyCurriculum
Press,lnc
Do notduplicata
withoutpsrmission
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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.
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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
=2,1 |
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withoutpormissbn.
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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.
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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
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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
freethrows
Prass,Inc.
o1S2 by KoyCurriculum
withoutpemission.
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I
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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 +
?(
5y = - 4
5y = 1 6
I
@199by KsyCuriculumPrsss,Inc.
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wilhoutp€rmission.
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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
x Y = 1 2
5x* y = 0
oppofteJ
(T".
vt-,r--l:r.-:
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
v
10
@1902
by KsyCurriculum
Prsss,Inc.
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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
The y-terns orc
thc sanc. If I
sb{rcct I1l g:t O.
11
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@19Cby KsyCurriculum
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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
x = y-3
/
= -tlx
5r*3I= l
6x*y = 6
Y = 2 ?(
5x+4/=5
v= 5 x - 1
12
x
= Y+
5
ol$? by KEyCuniculum
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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
13
6'199 by KeyCurriculum
Press,Inc,
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Solveeachsystemof equations
bysubstitution.
3x-l=7
= 2x - t+
I
y=@
2 x- f f =z
2x-(3+rl=2
2r-3 - * =I
x-3 =/
r = 5
l= !+ 5
g
Y=
Check:
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
Y = x - 1
2x-3y=ll
,l=5Ox+t-[
y - 32x = LiO
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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 *
y
x= 1 5
2x * 5,1= 2 6
15
@199 by Koy CurriculumPrsss,Inc
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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
x + ,f =-t+
-x *2Y = 13
x + y = 20
30r+25/ = 550
- 2 y =6
2x + Y =7
x
16
Prsss,Inc
@1992
by KeyCurriculum
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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
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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
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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:
19
O19e bt Ket CurdcubmPrss, Inc.
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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
=1260
60x*75,1
20
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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
21
Press,Inc.
@199by KeyCutriculum
Do notdupli:atswilhoutp€rmisshn.
www.pdfgrip.com
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
b
,r,,+ 2y = 2l
1r+Z+y=243
? ( + l = 2 0
60x*75y=1260
22
@1S2by K€yCurriculum
Press,Inc.
Do notduplicate
withoutp€rmission.
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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
23
@1992
by KeyCurriculum
Pross,Inc.
Do nol duplbalswithoutpermission.
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