Chapter 4: Systems of Equations and Inequalities 4... · Systems of Equations and Inequalities ... in 3, 4 or more variables. In this lesson we will learn to solve such systems algebraically.

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.