chapter 4: systems of equations and inequalities 4... · systems of equations and inequalities ......
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Chapter4:
SystemsofEquationsandInequalities
4.1SystemsofEquationsAsystemoftwolinearequationsintwovariablesxandyconsistoftwoequationsofthefollowingform:Equation1:𝑎𝑥 + 𝑏𝑦 = 𝑐 Equation2:𝑑𝑥 + 𝑒𝑦 = 𝑓 wherethesolution(x,y)satisfiesbothequations.CheckingSolutionsofaLinearSystem:3x–2y=2x+2y=6
1.) Is(2,2)asolutionoftheabovesystemofequations?
2.) Is(0,-1)asolutionoftheabovesystemofequations?SolvingaSystemGraphically:
Examples:1.)Solvethefollowingsystemofequationsgraphically.Determinehowmanysolutions.IdentifythesystemasConsistent,DependentConsistent,orInconsistent.Verifyyouransweronyourgraphingcalculator.2x–2y=-82x+2y=4 CheckAlgebraically..2.)Solvethefollowingsystemofequationsgraphically.Determinehowmanysolutions.IdentifythesystemasConsistent,DependentConsistent,orInconsistent.Verifyyouransweronyourgraphingcalculator.3x–2y=63x–2y=23.)Solvethefollowingsystemofequationsgraphically.Determinehowmanysolutions.IdentifythesystemasConsistent,DependentConsistent,orInconsistent.Verifyyouransweronyourgraphingcalculator.2x–2y=-8-2x+2y=8
SolvingaSystembySubstitutionSolveeachsystembelowbythemethodofSubstitution.
1) 2) SolvethesystembelowbythemethodofSubstitution,demonstratingthatthereisnosolution.
3) Whatdoesthegraphofthissystemlooklike?SolvethesystembelowbythemethodofSubstitution,demonstratingthatthereareinfinitelymanysolutions.
4) Whatdoesthegraphofthissystemlooklike?
y = 3x − 3y = −x + 5
−x + y = 33x + y = −1
2x − 2y = 0x − y = 1
x + y = 72x = 14 − 2y
ApplicationsUseyourgraphingcalculatortographthesystemofequationsforeachapplicationbelowandtoanswerrelatedquestions.Createasketchofeachgraph,labelingtheaxeswithappropriatescales.
1) YouarecheckingoutcellphoneplansanddiscoverthatTalkAnytimeWirelesscharges$50.00permonthforthefirstphonelineandcharges$20.00peradditionalphoneline.TextAwayWirelesscharges$80.00permonthforthefirstphonelineand$5.00peradditionalphoneline.DeterminethenumberofadditionalphonelinesforwhichitwouldbecheapertouseTalkAnytimeversesTextAway.
2) JamesandZachbegansavingmoneyfromtheirpart-timejobs.Jamesstartedwith$50inhissavingsandearns$10perhourathisjob.Zachstartedwith$225inhissavingsandearns$7.50perhour.Ifbothboyssavealloftheirearnings(andwedisregardtax)whenwilltheyhavethesameamountofsavings?
3) Youarechoosingbetweentwomovierentalservices.CompanyAcharges$2.99permovieplusa$20monthlyfee.CompanyBcharges$4.99permoviewithnomonthlyfee.Howmanymoviescouldyourentandgetchargedthesamemonthlybill?Ifyouonlyrent,onaverage,8moviespermonth,whichisthebetterdealforyou?
CheckforUnderstanding…
1) Youarecheckingasolutionofasystemoflinearequations.Howcanyoutellifthesolutionisvalidornot?
2) Describehowthegraphofasystemoflinearequationslookswhen…a. Thereisnotsolution.
b. Thereisexactlyonesolution.
c. Thereareinfinitelymanysolutions.
4.2LinearSystemsinTwoVariablesSolvingasystembythemethodofElimination
1.
2. 3.
3x + 2y = 45x − 2y = 8
4x − 5y = 133x − y = 7
3x + 9y = 82x + 6y = 7
Applications
1. Abusstation15milesfromtheairportrunsashuttleservicetoandfromtheairport.The9:00a.m.busleavesfortheairporttraveling30mph.The9:05a.m.busleavesfortheairporttraveling40mph.Writeasystemoflinearequationstorepresentdistanceasafunctionoftimeforeachbus.Howfarfromtheairportwillthe9:05a.m.buscatchuptothe9:00a.m.bus?
D = 30t
D = 40 t − 560
⎛⎝⎜
⎞⎠⎟
2. Theschoolyearbookstaffispurchasingadigitalcamera.Recentlythestaffreceivedtwoadsinthemail.Theadforstore#1statesthatalldigitalcamerasare15%off.Theadforstore#2givesa$300coupontousewhenpurchasinganydigitalcamera.Assumethatthelowestpriceddigitalcamerais$700.Whencouldyougetthesamedealateitherstore?
LetC=thecostofacameraafterthediscount
Letx=theoriginalcostofacamera
3. Youarestartingabusinesssellingboxesofhand-paintedgreetingcards.Togetstarted,youspend$36onpaintandpaintbrushesthatyouneed.Youbuyboxesofplaincardsfor$3.50perbox,paintthecards,andthensellthemfor$5perbox.Howmanyboxesmustyousellforyourearningstoequalyourexpenses?Whatwillyourearningsandexpensesequalwhenyoubreakeven?(WriteanequationtorepresentTotalExpensesandanotherequationtorepresentTotalEarnings.)
4. Youcommutetocentercity5daysperweekonaSEPTAtrain.Youcanpurchaseamonthlypassfor$140permonthorpurchasearoundtripticketeachdaythatyoucommutefor$9.50perticket.Whatisthenumberofdaysthatyoumustridetobeginsavingmoneybyusingthemonthlypass?
C=thecostin$
x=thenumberofdayscommuting
5. Asoccerleagueofferstwooptionsformembershipplans.OptionA:aninitialfeeof$40andthenyoupay$5foreachgamethatyouplay.OptionB:youhavenoinitialfeebutmustpay$10foreachgamethatyouplay.Afterhowmanygameswillthetotalcostofthetwooptionsbethesame?
4.3LinearSystemsinThreeVariablesInadditiontosystemsoftwoequations,itissometimesnecessarytosolveasystemof3,4ormoreequationsin3,4ormorevariables.Inthislessonwewilllearntosolvesuchsystemsalgebraically.Laterinthechapterwewilluseamatrixandourgraphingcalculatortosolvesuchsystems.BackSubstitutionThisexamplehasareasonablystraightforwardsetupallowingustousesimplebacksubstitutiontosolve.
MethodofEliminationThisexamplerequiresthatweeliminatexbycombiningEquations1and2,andalsoeliminatexbycombiningEquations2and3.WecannowusetheEliminationmethodtosolvetheresultingequationsforyandz,andthenbacksubstitutetosolveforx.Example1:
x − 2y + 2z = 9y + 2z = 5z = 3
x − 2y + 2z = 9 Equation 1−x + 3y = 4 Equation 2
2x − 5y + z = 10 Equation 3
Dependingonthesetupofthesystem,youmaywishtoeliminateyorzfromtheoriginalpairsofequations.Example2:
Howmanysolutionsarepossible??Thegraphofasystemof3linearequationsin3variablesconsistsof3planes.Theplanesmayintersectinonepoint,inoneline,inoneplaneornotatall.
4x + y − 3z = 112x − 3y + 2z = 9x + y + z = −3
AnInconsistentSystem:
ASystemwithInfinitelyManySolutions:
x − 3y + z = 12x − y − 2z = 2x + 2y − 3z = −1
x + y − 3z = −1y − z = 0−x + 2y = 1
Let’slookattheseapplicationsfromyourtextbook.
6 2 −1−2 0 5
⎡
⎣⎢
⎤
⎦⎥
4.4MatricesandLinearSystemsand4.5DeterminantsandLinearSystems(Day1)MatrixOperationsAlwaysreadamatrixROWbyCOLUMN #Rows:________ Dimension:________ #Columns:_____ Numbersinthematrixarecalledentries.Whatistheentryinthe2ndrowand3rdcolumnforthematrixabove?
DifferentTypesofMatrices
Name Example DimensionsRowMatrix 1 −7 0 5⎡⎣ ⎤⎦ 1x4
ColumnMatrix 8710
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥ 3x1
SquareMatrix −2 3 510 1 17 13 22
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥ 3x3
Whatarethedimensionsofeachmatrixbelow?
1 15 2 −7 314 8 12 0 0−2 4 3 7 10
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥ 6 12 9 −2 1⎡⎣ ⎤⎦
3 612 719 234 8
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
MatrixAdditionandSubtraction:§ MatricesmusthavetheSAMEdimensions.§ Addorsubtractthecorrespondingentries.
Example1: 6 5 − 2−3 − 2 0⎡
⎣⎢
⎤
⎦⎥ +
−10 8 13−9 1 − 7⎡
⎣⎢
⎤
⎦⎥
Dimensionofeachmatrix:________ Dimensionoftheanswermatrix:_______
Example2: 8 34 0
⎡
⎣⎢
⎤
⎦⎥ −
2 −76 −1
⎡
⎣⎢
⎤
⎦⎥ =
Dimensionofeachmatrix:________ Dimensionoftheanswermatrix:_______ScalarMultiplication:
§ MultiplytheconstantOUTSIDEthematrixtoEACHentryinsidethematrix.
Example3: 3 −2 04 −7
⎡
⎣⎢
⎤
⎦⎥ =
Dimensionoftheanswermatrix:_________ScalarMultiplicationcombinedwithAdditionorSubtraction:
Example4: −21 −20 3−4 5
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥+
−4 56 −8−2 6
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=
Dimensionofeachmatrix:________ Dimensionoftheanswermatrix:_______Solvethefollowingmatrixforxandy
§ Correspondingentriesareequal
Example5: 2 3x −18 5
⎡
⎣⎢
⎤
⎦⎥ +
4 1−2 −y
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎛
⎝⎜
⎞
⎠⎟ =
26 012 8
⎡
⎣⎢
⎤
⎦⎥
4.4,4.5HomeworkDay1:MatrixoperationsPerformtheindicatedoperationifpossible.Ifnotpossible,statethereason.
1. 2.
3. 4. Solvethematrixequationforxandy.
5.
6.
7.
8.
15 43 12
⎡
⎣⎢
⎤
⎦⎥ −
0 92 7
⎡
⎣⎢
⎤
⎦⎥ =
3 −2−4 1
⎡
⎣⎢
⎤
⎦⎥ −
5−3
⎡
⎣⎢
⎤
⎦⎥ =
6 109 64 −1
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥+
2 10 74 7
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥= 2
4 6 −110 −5 20 11 1
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=
1 14−5x 10
⎡
⎣⎢
⎤
⎦⎥ =
y − 9 145 10
⎡
⎣⎢⎢
⎤
⎦⎥⎥
3 4y−1 13
⎡
⎣⎢⎢
⎤
⎦⎥⎥+ −6 5
8 0⎡
⎣⎢
⎤
⎦⎥ =
−3 −7x 13
⎡
⎣⎢
⎤
⎦⎥
2 3y4 −1
⎡
⎣⎢⎢
⎤
⎦⎥⎥+ 0 −4
x −2⎡
⎣⎢
⎤
⎦⎥ =
2 113 −3
⎡
⎣⎢
⎤
⎦⎥
7y −2−3 5
⎡
⎣⎢⎢
⎤
⎦⎥⎥− 1 5
x −3⎡
⎣⎢
⎤
⎦⎥ =
6 −7−2 8
⎡
⎣⎢
⎤
⎦⎥
(Day2)MatrixMultiplication:
§ Thenumberofcolumnsinthefirstmatrixmustmatchthenumberofrowsinthesecondmatrix.If[A]hasdimensionsmxn If[B]hasdimensionsnxp Theproductof[A]x[B]willhavedimensionsmxpA:2X3 B:3X4 A:3X2 B:3X4Dimensionof[A]x[B]:________ Dimensionof[A]x[B]:________Example6: FindAB
A =−2 31 −46 0
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
B = −1 3−2 4
⎡
⎣⎢
⎤
⎦⎥
Dimof[A]:_________ Dimof[B]:_________ ProductDim:______________Example7: FindBA
A =−2 31 −46 0
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
B = −1 3−2 4
⎡
⎣⎢
⎤
⎦⎥
Dimof[A]:_________ Dimof[B]:_________ ProductDim:______________Example8: FindAB+BC
A = 2 1−1 3
⎡
⎣⎢
⎤
⎦⎥, B = −2 0
4 2⎡
⎣⎢
⎤
⎦⎥, and C = 1 1
3 2⎡
⎣⎢
⎤
⎦⎥
Useyourcalculatortoadd,subtract,multiplywithmatrices.ToenteraMatrixinyourcalculator:2ndMATRIX EDITENTER(enterthedimensionsofthematrixandtheentries)TocallupaMatrixinyourcalculatorfromthehomescreen:2ndMATRIX(highlightthematrix)ENTER
A = 2 1−1 3
⎡
⎣⎢
⎤
⎦⎥, B = −2 0
4 2⎡
⎣⎢
⎤
⎦⎥, and C = 1 1
3 2⎡
⎣⎢
⎤
⎦⎥
1.)B(A+C) 2.)BA+BCApplicationofMatrices:Ahealthcluboffersthreedifferentmembershipplans.WithPlanX,youcanuseallclubfacilities:thepool,fitnesscenter,andracketclub.WithPlanY,youcanusethepoolandfitnesscenter.WithPlanZ,youcanonlyusetheracketclubfacilities.ThematricesbelowshowtheannualcostforaSingleandaFamilymembershipfortheyears2012through2014. [A] [B] [C]
2012 2013 2014 singlefamily singlefamily singlefamily
plan Xplan Yplan Z
336 624228 528216 385
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
plan Xplan Yplan Z
384 720312 576240 432
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
plan Xplan Yplan Z
420 792360 672288 528
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
1)Determineamatrixthatgivesthepriceincreasefrom2012to2014foreachoftheplans.2)Determineamatrixthatgivesthetotalcostforallthreeyearsforeachoftheplans.3)Thehealthcluboffereda3-yearmembershipbasedonthe2012rates.Howmuchmoneydoesthe3-yearmembershipsaveforeachplancomparedtopayingtheregularmembershiprateforeachofthe3years?
Homework4.4,4.5Day2:MultiplyingmatricesForthematriceswiththegivendimensions,whatarethedimensionsoftheproduct?Iftheproductisundefined,explainwhy.1. A:2X5 B:5X3 2. A:6X2 B:3X1Dimensionof[A]x[B]:________ Dimensionof[A]x[B]:________3. A:3X1 B:1X2 4. A:1X6 B:6X1Dimensionof[A]x[B]:________ Dimensionof[A]x[B]:________Writetheproduct.Ifitisnotdefined,statethereason.
5. 6.
7. 8. GivenmatricesA,BandC,determinetheproducts.Iftheproductisnotdefined,statethereason.
9. [A][B]= 10. [A][C]= 11. [C][B]=12. [B][C]= 13. [C][A]= 14. [B][A]=
12−4
⎡
⎣⎢
⎤
⎦⎥ −10 −7⎡⎣ ⎤⎦ =
2 15−3 10
⎡
⎣⎢
⎤
⎦⎥
−5 121 0
⎡
⎣⎢
⎤
⎦⎥ =
1 70 9
⎡
⎣⎢
⎤
⎦⎥
3 −1 82 −4 8
⎡
⎣⎢
⎤
⎦⎥ =
−3 2 12−1 0 5
⎡
⎣⎢
⎤
⎦⎥
3 4−7 15
⎡
⎣⎢
⎤
⎦⎥ =
A = 3 2−7 5
⎡
⎣⎢
⎤
⎦⎥
B =256
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
C = 0 5 −32 1 6
⎡
⎣⎢
⎤
⎦⎥
(Day 3) Useamatrixandagraphingcalculatortosolvealinearsystem2ndMATRIXEDITENTER(editmatrix)2ndMATRIXMATHB↓ rref(2ndMATRIX(selectthematrixthatyouedited)
1)−2x − y + 4z = −48−x + 2y + 2z = 6x − 3y + 4z = −54
Usethematrix:−2 −1 4 −48−1 2 2 61 −3 4 −54
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
Solutionmatrix:1 0 0 x0 1 0 y0 0 1 z
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
2)x + y − 2z = −92x + y + z = 0−x − 2y + 6z = 21
Homework4.4,4.5Day3:UseamatrixtosolveasystemSolvethesystemofequationsusingamatrix.1. 9x+8y=-6 -x–y=12. x–3y=-2 5x+3y=173. x–y–4z=3 -x+3y–z=-1 x–y+5z=34. 4x+10y–z=-3 11x+28y–4z=1 -6x–15y+2z=-15. 5x–3y+5z=-1 3x+2y+4z=11 2x–y+3z=4
(Day 4) Determinants Determinantofa2x2matrix:
detdcba
dcba
=⎥⎦
⎤⎢⎣
⎡ =ad–bc
3 4−2 8
12
43
35
710
Determinantofa3x3matrix:
deta b cd e fg h i
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
= (aei+bfg+cdh)–(ceg+afh+bdi)
4 3 15 − 7 01 − 2 2
5 2 −34 −1 76 1 2
Evaluateadeterminantinyourcalculator:1) Enter the determinant as a matrix: 2nd MATRIX EDIT ENTER 2nd QUIT (enterthedimensionsofthematrixandtheentries)2) Evaluate the determinant: 2nd MATRIX MATH 1:det( 2nd MATRIX Select the matrix that
you edited. ENTER Checkthevalueofthedeterminantsabovebyusingyourcalculator.
Homework4.4,4.5Day4:DeterminantsandsystemsEvaluatethedeterminant.
1. −3 47 5
= 2. 12 −14 2
=
UseMatricestosolvethesystemofequations.
1.x + y − z = −32x − 3y + 4z = 23−3x + y − 2z = −15
2.3x + 3y + 4z = 13x + 5y + 9z = 25x + 9y +17z = 4
3.5x + 3y − 2z = −42x + 2y + 2z = 03x + 2y +1z = 1
4.2x − 4y + 5z = −334x − y = −5−2x + 2y − 3z = 19
Applications:1. ClaireandDaleshoppedatthesamestore.Clairebought5kgofapplesand2kgofbananasandpaidaltogether$22.Dalebought4kgofapplesand6kgofbananasandpaidaltogether$33.Usematricestofindthecostof1kgofbananas.
2. AnnandBillybothenteredaquiz.Thequizhadtwentyquestionsandpointswereallocatedasfollows:§ P points were added for each correctly answered question § Q points were deducted for each incorrect (or unanswered) question
Anngot15questionscorrectandscored65points.Billygot11questionscorrectandscored37points.UsematricestofindthevalueofQ.3. Acommunityrelieffundreceivesalargedonationof$2800.Thefoundationagreestospendthemoneyon$20schoolbags,$25sweaters,and$5notebooks.Theywanttobuy200itemsandsendthemtoschoolsinearthquake-hitareas.Theymustorderasmanynotebooksasschoolbagsandsweaterscombined.Howmanyofeachitemshouldtheyorder?4. AnultimateFrisbeeteamhastoorderjerseys,shorts,andhats.Theyhaveabudgetof$1350tospendon$50jerseys,$20shorts,and$15hats.Theywanttobuy40itemsinpreparationfortheoncomingseasonandmustorderasmanyjerseysasshortsandhatscombined.Howmanyofeachitemshouldtheyorder?
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
4.6SystemsofLinearInequalities
Graphthefollowingsystemsofinequalitiesandlabelthevertex/vertices:
1)
y ≥ −3x −1y < x + 2
2.) x ≤ 0y ≥ 0x − y ≥ −2
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
3.)−x < yx + 3y < 9x ≥ 2
4.)x + 2y ≤ 102x + y ≤ 82x − 5y < 20
Writethesystemofinequalitiesthatcorrespondwiththeshadedregion.
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
4.6HOMEWORKGraphthesystemoflinearinequalities.
1)
�
y > −2y ≤ 1
2)
�
y > −5xx ≤ 5y
3)
�
x − y > 72x + y < 8
4)
�
y < 4x > −3y > x
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
10
8
6
4
2
–2
–4
–6
–8
–10
–10 –5 5 10
5)
�
2x − 3y > −65x − 3y < 3x + 3y > −3
6)
�
y < 5y > −62x + y ≥ −1y ≤ x + 3
Challenge.Writeasystemoflinearinequalitiesfortheregion.
ReviewWorksheetforChapter4TestCompletethefollowingproblemsfromthee-book:p.290-293(9,11,15,17,23,27,31,33,37,39,41,43,71)Completethefollowingproblemswithmatrices.