Mathematics - Gearing Towards Quality Education · 08-06-2012 · Mathematics Learner’s Module 5 Department of Education Republic of the Philippines This instructional material

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Mathematics Learner’s Module 5

Department of Education Republic of the Philippines

This instructional material was collaboratively developed and

reviewed by educators from public and private schools,

colleges, and/or universities. We encourage teachers and

other education stakeholders to email their feedback,

comments, and recommendations to the Department of

Education at [email protected].

We value your feedback and recommendations.

Mathematics – Grade 8 Learner’s Module First Edition, 2013

ISBN: 978-971-9990-70-3

Republic Act 8293, section 176 indicates that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may among other things, impose as a condition the payment of royalties.

The borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Br. Armin Luistro FSC Undersecretary: Dr. Yolanda S. Quijano

Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 2nd Floor Dorm G, PSC Complex, Meralco Avenue.

Pasig City, Philippines 1600 Telefax: (02) 634-1054, 634-1072 E-mail Address: [email protected]

Development Team of the Learner’s Module Consultant: Maxima J. Acelajado, Ph.D.

Authors: Emmanuel P. Abuzo, Merden L. Bryant, Jem Boy B. Cabrella, Belen P. Caldez, Melvin M. Callanta, Anastacia Proserfina l. Castro, Alicia R. Halabaso, Sonia P. Javier, Roger T. Nocom, and Concepcion S. Ternida

Editor: Maxima J. Acelajado, Ph.D.

Reviewers: Leonides Bulalayao, Dave Anthony Galicha, Joel C. Garcia, Roselle Lazaro, Melita M. Navarro, Maria Theresa O. Redondo, Dianne R. Requiza, and Mary Jean L. Siapno

Illustrator: Aleneil George T. Aranas

Layout Artist: Darwin M. Concha

Management and Specialists: Lolita M. Andrada, Jose D. Tuguinayo, Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel, Jr.

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Table of Contents Unit 2

Module 5: Systems of Linear Equations and

Inequalities in Two Variables .................................................. 243

Module Map ....................................................................................................... 244

Pre-Assessment ................................................................................................ 245

Learning Goals .................................................................................................. 252

Lesson 1: Rational Algebraic Expressions .................................................... 253

Activity 1 ........................................................................................................ 253

Activity 2 ........................................................................................................ 254

Activity 3 ........................................................................................................ 258

Activity 4 ........................................................................................................ 259

Activity 5 ........................................................................................................ 260

Activity 6 ........................................................................................................ 263

Activity 7 ........................................................................................................ 264

Summary/Synthesis/Generalization ............................................................... 267

Lesson 2: Solving Systems of Linear Equations in Two Variables .............. 268

Activity 1 ........................................................................................................ 268

Activity 2 ........................................................................................................ 270

Activity 3 ........................................................................................................ 271

Activity 4 ........................................................................................................ 272

Activity 5 ........................................................................................................ 278

Activity 6 ........................................................................................................ 279

Activity 7 ........................................................................................................ 280

Activity 8 ........................................................................................................ 280

Activity 9 ........................................................................................................ 281

Activity 10 ...................................................................................................... 283

Activity 11 ...................................................................................................... 284

Activity 12 ...................................................................................................... 284

Activity 13 ...................................................................................................... 285

Activity 14 ...................................................................................................... 286

Activity 15 ...................................................................................................... 286


Lesson 3: Graphical Solutions of Systems of Linear Inequalities

in Two Variables .............................................................................................. 290

Activity 1 ........................................................................................................ 290

Activity 2 ........................................................................................................ 292

Activity 3 ........................................................................................................ 295

Activity 4 ........................................................................................................ 296

Activity 5 ........................................................................................................ 297

Activity 6 ........................................................................................................ 299

Activity 7 ........................................................................................................ 300

Activity 8 ........................................................................................................ 302

Activity 9 ........................................................................................................ 302


Glossary of Terms ........................................................................................... 305

References and Website Links Used in this Module ..................................... 306

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I. INTRODUCTION AND FOCUS QUESTIONS Haveyoueveraskedyourselfhowbusinessmenmakeprofits?Howcanfarmersincreasetheiryieldorharvest?Howparentsbudgettheir incomeonfood,education,clothingandotherneeds?Howcellularphoneuserschoosethebestpaymentplan?Howstudentsspendtheirdailyallowancesortravelfromhometoschool?

Findouttheanswerstothesequestionsanddeterminethevastapplicationsofsystemsoflinearequationsandinequalitiesintwovariablesthroughthismodule.

II. LESSONS AND COVERAGE

Inthismodule,youwillexaminetheabovequestionswhenyoutakethefollowinglessons:

Lesson 1–SystemsoflinearequationsintwovariablesandtheirgraphsLesson 2–SolvingsystemsoflinearequationsintwovariablesLesson 3–Graphicalsolutionsofsystemsoflinearinequalitiesintwovariables

SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES

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Intheselessons,youwilllearnto:Lesson 1 • Describe systems of linear equations and inequalities using

practicalsituationsandmathematicalexpressions.• Identifywhichsystemsof linearequationshavegraphs thatare

parallel,intersecting,andcoinciding.• Graphsystemsoflinearequationsintwovariables.

Lesson 2 • Solvesystemsoflinearequationsby(a)graphing;(b)elimination;(c)substitution.

• Solve problems involving systems of linear equations in twovariables.

Lesson 3 • Graphasystemoflinearinequalitiesintwovariables.• Solveasystemoflinearinequalitiesintwovariablesbygraphing.• Solve problems involving systems of linear inequalities in two

variables.

Module MapModule Map Hereisasimplemapofthelessonsthatwillbecoveredinthismodule.

SystemsofLinearEquationsandInequalitiesin

Two Variables

SystemsofLinearEquationsinTwoVariablesandtheir

Graphs

SolvingSystemsofLinearEquationsin

Two Variables

GraphicalSolutionsofSystemsofLinearInequalitiesinTwo

Variables

GraphicalMethod

EliminationMethod

AlgebraicMethods

SubstitutionMethod

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III. PRE - ASSESSMENT

Part I: Findouthowmuchyoualreadyknowaboutthismodule.Choosetheletterthatyouthinkbestanswersthequestion.Pleaseanswerallitems.Takenoteoftheitemsthatyouwerenotabletoanswercorrectlyandfindtherightanswerasyougothroughthismodule.

1. Whichofthefollowingisasystemoflinearequationsintwovariables?

a. 2x – 7y=8 c. x + 9y = 22x – 3y > 12

b. 3x + 5y = -2x – 4y = 9 d. 4x + 1 = 8

2. Whatpointistheintersectionofthegraphsofthelinesx + y = 8 and 2x – y = 1?

a. (1,8) b. (3,5) c. (5,3) d. (2,6)

3. Whichofthefollowingisagraphofasystemoflinearinequalitiesintwovariables?

a. c.

b. d.

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4. Whichofthefollowingshowsthegraphofthesystem2x + y < 2x – 4y > 9 ?

a. c.

b. d.

5. If2x + y = 9 and 2x – y=11,whatisthevalueofx?

a. 4 b. 5 c. 10 d. 20

6. AcarparkchargesPhp45forthefirst3hoursandPhp5foreverysucceedinghourorafractionthereof.AnothercarparkchargesPhp20forthefirst3hoursandPhp10foreverysucceedinghourorafractionthereof.Inhowmanyhourswouldacarownerpaythesameparkingfeeinanyofthetwocarparks?

a. 2hr b. 3hr c. 5hr d. 8hr

7. Howmanysolutionsdoesaconsistentandindependentsystemoflinearequationshave?

a. 0 b. 1 c. 2 d. Infinite

8. Whichofthefollowingorderedpairssatisfyboth2x + 7y > 5 and 3x – y≤2?

a. (0,0) b. (10,-1) c. (-4,6) d. (-2,-8)

9. Mr.AgpalopaidPhp260for4adult’sticketsand6children’stickets.Supposethetotalcostofanadult'sticketandachildren’sticketisPhp55.Howmuchdoesanadult'sticketcost?

a. Php20 b. Php35 c. Php80 d. Php120

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10. Whichsystemofequationshasagraphthatshowsintersectinglines? a. 2x + 4y = 14

x + 2y = 7 c. 4x + 8y = 7

x + 2y = 3

b. -3x + y = 5 6x – 2y = 1

d. 3x + y=103x – y = 5

11. Mr. Bonifacio asked each of his agriculture students to prepare a rectangulargardensuch that itsperimeter isatmost19mand thedifferencebetween itslengthanditswidthisatleast5m.Whichofthefollowingcouldbethesketchofagardenthatastudentmayprepare?

a. c.

b. d.

12. Luisasaysthatthesystem3x + y = 2 2y = 15 –6xhasnosolution.Whichofthefollowing

reasonswouldsupportherstatement? I. Thegraphofthesystemofequationsshowsparallellines. II. Thegraphofthesystemofequationsshowsintersectinglines. III. Thetwolinesasdescribedbytheequationsinthesystemhavethesame

slope.

a. IandII b. IandIII c. IIandIII d. I,II,andIII

13. JosepaidatmostPhp250forthe4markersand3pencilsthathebought.Supposethemarkerismoreexpensivethanthepencilandtheirprice’sdifferenceisgreaterthanPhp30.WhichofthefollowingcouldbetheamountpaidbyJoseforeachitem?

a. Marker: Php56 c. Marker: Php46 Pencil: Php12 Pencil: Php7 b. Marker: Php35 d. Marker: Php50 Pencil: Php15 Pencil: Php19

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14. Bea wanted to compare the mobile network plans being offered by twotelecommunicationcompanies.SupposeBea’sfatherwouldliketoseethegraphshowingthecomparisonofthetwomobilenetworkplans.WhichofthefollowinggraphsshouldBeapresenttohisfather?

a. c.

b. d.

15. EdnaandGracehadtheirmealatapizzahouse.Theyorderedthesamekindofpizzaanddrinks.EdnapaidPhp140for2slicesofpizzaandadrink.GracepaidforPhp225for3slicesofpizzaand2drinks.Howmuchdidtheypayforthetotalnumberofslicesofpizza?

a. Php55 b. Php110 c.Php165 d.Php275 16. The Senior Citizens' Club of a certain municipality is raising funds by selling

used clothes and shoes.Mrs. Labrador, amember of the club, was assignedto determine howmany used clothes and shoes were sold after knowing theimportantinformationneeded.Shewasaskedfurthertopresenttotheclubhowshecameupwiththeresultusingagraph.WhichofthefollowinggraphscouldMrs.Labradorpresent?

a. c.

b. d.

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17. TheMathClubrentedasoundsystemfortheirannualMathematicsCamp.Theyalsorentedageneratorincaseofpowerinterruption.Afterthe3-daycamp,theclubpaidatotalamountofPhp3,000,threedaysforthesoundsystemandtwodaysforthegenerator.Ifeachisrentedforoneday,theclubshouldhavepaidatotalamountofPhp1,100.Whatwasthedailyrentalcostofthegenerator?

a. Php300 c. Php800 b. Php600 d. Php2,400

18. Mrs.Sorianowould liketokeeptrackofher family’sexpensestohavean ideaofthemaximumorminimumamountofmoneythatshewillallotforelectricandwaterconsumption,food,clothing,andotherneeds.WhichofthefollowingshouldMrs.Sorianoprepare?

a. BudgetPlan c. PricelistofCommodities b. CompilationofReceipts d. BarGraphofFamily’sExpenses

19. A restaurantownerwould like tomakeamodelwhichhecanuseasguide inwritingasystemofequations.Hewillusethesystemofequationsindeterminingthenumberofkilogramsofporkandbeefthatheneedstopurchasedailygivenacertainamountofmoney(C),thecost(A)ofakiloofpork,thecost(B)ofakiloofbeef,andthetotalweightofmeat(D).Whichofthefollowingmodelsshouldhemakeandfollow?

a. Ax – By = C x + y = D c. Ax + By = C

x + y = D

b. Ax + By = C x – y = D d. Ax – By = C

x – y = D

20. Mrs.Jacintowouldliketoinstillthevalueofsavingandtodevelopdecision-makingamongherchildren.WhichofthefollowingsituationsshouldMrs.Jacintopresenttoherchildren?

a. Buyingandsellingdifferentitems b. Apersonputtingcoinsinhispiggybank c. Buyingassortedgoodsinadepartmentstore d. Makingbankdepositsintwobanksthatgivedifferentinterests

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Part II.Illustrate each mathematics concept in the given figure, then describe it bycompletingthestatementatthebottom.

Myideaof(mathematicsconceptgiven)is____________________________________________________________________________________________________________________________________________________________________________________________

Lines

Points

Slope of a Line

Points on a Line

ParallelLines

IntersectingLines

Linear Inequality

y - intercept of a Line

Coordinates of Points

Linear Equations

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Part III. Usethesituationbelowtoanswerthequestionsthatfollow.

OneSunday,aButterflyExhibitwasheldattheQuezonMemorialCircleinQuezonCity.Anumberofpeople,childrenandadults,wenttoseetheexhibit.AdmissionwasPhp20eachforadultsandPhp12eachforchildren.

Questions:

1. Howmuchdidanadultpayfortheexhibit?Howaboutachild?

2. Completethetablebelowfortheamountthatmustbepaidbyacertainnumberofadultsandchildrenwhowillwatchtheexhibit.

Number of Adults Admission Fee Number of

Children Admission Fee

2 23 34 45 56 6

3. Howmuchwould10adultspayiftheywatchtheexhibit?Howabout10children?Showyoursolution.

4. Ifacertainnumberofadultswatchedtheexhibit,whatexpressionwouldrepresentthetotaladmissionfee?

What mathematical statement would represent the total amount that will becollectedfromanumberofchildren?Explainyouranswer.

5. Suppose6adultsand15childrenwatchtheexhibit.Whatisthetotalamounttheywillpayasadmission?Showyoursolution.

6. Ifanumberofadultsandanothernumberofchildrenwatchtheexhibit,howwillyou represent the total amount they will pay for the admission? Explain youranswer.

7. SupposethetotalamountcollectedwasPhp3,000.Howmanyadultsandhowmanychildrencouldhavewatchedtheexhibit?

8. Thegivensituationillustratestheuseoflinearequationsintwovariables.Inwhatotherreal-lifesituationsarelinearequationsintwovariablesapplied?Formulateproblemsoutofthesesituationsthensolve.

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IV. LEARNING GOALS AND TARGETS

Aftergoingthroughthismodule,youshouldbeabletodemonstrateunderstand-ingofkeyconceptsofsystemsof linearequationsand inequalities in twovariables,formulatereal-lifeproblemsinvolvingtheseconcepts,andsolvethesewithutmostac-curacyusingavarietyofstrategies.

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DESCRIBE ME!Activity 1

Directions: Draw the graph of each of the following linear equations in a Cartesiancoordinateplane.Answerthequestionsthatfollow.

1. y = 2x+3 2.3x – y = 2

11Systems of Linear Equations in Two

Variables and Their Graphs

Lesson

What to KnowWhat to Know

Start Lesson 1 of this module by assessing your knowledge of the differentmathematics concepts previously studied and your skills in performing mathematicaloperations.TheseknowledgeandskillsmayhelpyouinunderstandingSystemsofLinearEquationsinTwoVariablesandtheirGraphs.Asyougothroughthislesson,thinkofthefollowingimportantquestion:“How is the system of linear equations in two variables used in solving real-life problems and in making decisions?”Tofindtheanswer,performeachactivity.Ifyoufindanydifficultyinansweringtheexercises,seektheassistanceofyourteacherorpeersorrefertothemodulesyouhavestudiedearlier.Tocheckyourwork,refertotheanswerkeyprovidedattheendofthismodule.

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3. y = 5x–1 4.2x – 3y=6 Q

U

ESTIONS? a. Howdidyougrapheachlinearequationintwovariables?

b. Howdoyoudescribethegraphsoflinearequationsintwovariables?

Were you able to draw and describe the graphs of linear equations in two variables? Suppose you draw the graphs of two linear equations in the same coordinate plane. How would the graphs of these equations look like? You’ll find that out when you do the next activity.

MEET ME AT THIS POINT IF POSSIBLE…Activity 2

Directions: Draw the graph of each pair of linear equations below using the sameCartesianplane,thenanswerthequestionsthatfollow.

1. 3x + y = 5 and 2x + y=9 2. 3x – y = 4 and y = 3x + 2

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3. x + 3y=6and2x+6y = 12

QU

ESTIONS?

a. Howdidyougrapheachpairoflinearequations?b. Howwouldyoudescribethegraphsof3x + y = 5 and 2x + y=9? Howabout3x – y = 4 and y = 3x+2?x + 3y=6and2x+6y=12?c. Whichpairofequationshasgraphsthatareintersecting? Howmanypointsofintersectiondothegraphshave? Whatarethecoordinatesoftheirpoint(s)ofintersection?d. Whichpairofequationshasgraphsthatarenotintersecting?Why? Howdoyoudescribetheseequations?

e. Eachpairoflinearequationsformsasystemofequations.Thepointofintersectionofthegraphsoftwolinearequationsisthesolutionofthesystem.Howmanysolutionsdoeseachpairofequationshave?

e.1)3x + y = 5 and 2x + y = 9 e.2)3x – y = 4 and y = 3x + 2 e.3)x + 3y=6and2x+6y = 12f. Whatistheslopeandthey-interceptofeachlineinthegivenpairof

equations? f.1) 3x + y=5; slope= y-intercept= 2x + y=9; slope= y-intercept=

f.2) 3x – y=4; slope= y-intercept= y = 3x+2; slope= y-intercept=

f.3) x + 3y=6; slope= y-intercept= 2x+6y=12; slope= y-intercept=

g. Howwouldyoucomparetheslopesofthelinesdefinedbythelinearequationsineachsystem?

Howabouttheiry-intercepts? h. Whatstatementscanyoumakeaboutthesolutionofthesystemin

relationtotheslopesofthelines? Howaboutthey-interceptsofthelines?

i. Howisthesystemoflinearequationsintwovariablesusedinsolvingreal-lifeproblemsandinmakingdecisions?

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How did you find the preceding activities? Are you ready to learn about systems of linear equations in two variables and their graphs? I’m sure you are. From the activities done, you were able to determine when two lines intersect and when they do not intersect. You were able to relate also the solution of system of linear equations with the slopes and y-intercepts of their graphs. But how are systems of linear equations in two variables used in solving real-life problems and in making decisions? You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on Systems of Linear Equations in Two Variables and their Graphs and the examples presented.

Equationslikex – y = 7 and 2x + y=8arecalledsimultaneous linear equations or a system of linear equationsifwewantthemtobetrueforthesamepairsofnumbers.Asolutionofsuchequationsisanorderedpairofnumbersthatsatisfiesbothequations.Thesolutionsetofasystemoflinearequationsintwovariablesisthesetofallorderedpairsofrealnumbersthatmakeseveryequationinthesystemtrue.

The solution of a system of linear equations can be determined algebraically orgraphically.Tofindthesolutiongraphically,graphbothequationsonaCartesianplanethenfind the point of intersection of the graphs, if it exists. The solution to a system of linearequationscorresponds to thecoordinatesof thepointsof intersectionof thegraphsof theequations.

Asystemoflinearequationshas: a. onlyonesolutioniftheirgraphsintersectatonlyonepoint. b. nosolutioniftheirgraphsdonotintersect. c. infinitelymanysolutionsiftheirgraphscoincide.

Exactlyonesolution Nosolution Infinitelymanysolutions

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Therearethreekindsofsystemsoflinearequationsintwovariablesaccordingtothenumberofsolutions.Theseare: 1. System of consistent and dependent equations

Thisisasystemoflinearequationshavinginfinitelymanysolutions.Theslopesofthelinesdefinedbytheequationsareequal,theiry-interceptsarealsoequal,andtheirgraphscoincide.

Example: The system of equations

x – y = 5 2x – 2y =10 is consistent and

dependent. The slopes of theirlinesareequal, theiry-interceptsare also equal, and their graphscoincide.

2. System of consistent and independent equations Thisisasystemoflinearequationshavingexactlyonesolution.Theslopesofthe

linesdefinedbytheequationsarenotequal;theiry-interceptscouldbeequalorunequal;andtheirgraphsintersect.

Example: Thesystemof equations 2x + y = 5 3x – y = 9

is consistent and independent. The slopes of their lines are not equal,their y-intercepts could be equal orunequal,andtheirgraphsintersect.

3. System of inconsistent equations This isasystemof linearequationshavingnosolution.Theslopesofthelines

defined by the equations are equal, their y-intercepts are not equal; and theirgraphsareparallel.

Example: Thesystemofequations 2x + y=-62x + y =10isinconsistent.Theslopesof

theirlinesareequal;their y-interceptsarenotequal;andtheirgraphsareparallel.

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Systems of linear equationsin two variables are illustratedin many real-life situations. Asystem of linear equations intwo variables can be used torepresent problems that involvefinding values of two quantitiessuch as the number of objects,costs of goods or services, oramountofinvestments,solutionsof which can also be describedusing graphs. But how are thesolutions to problems involvingsystemsoflinearequationsusedinmakingdecisions?

What to ProcessWhat to Process

Yourgoalinthissectionistoapplythekeyconceptsofsystemsoflinearequationsintwovariablesandtheirgraphs.Usethemathematicalideasandtheexamplespresentedintheprecedingsectiontoanswertheactivitiesprovided.

CONSISTENT OR INCONSISTENT?Activity 3

Directions: Determine whether each system of linear equations is consistent anddependent,consistentandindependent,orinconsistent.Then,answerthequestionsthatfollow.

1. 2x – y = 73x – y = 5 6. x – 2y = 9

x + 3y = 14 2. 2x + y = -3

2x + y =6 7. 6x – 2y = 8y = 3x – 4

3. x – 2y = 92x – 4y = 18 8. x + 3y = 8

x – 3y = 8

4. 8x + 2y = 7y = -4y + 1 9. 2y=6x – 5

3y = 9x + 1

5. -3x + y=104x + y = 7 10. 3x + 5y = 15

4x – 7y =10

LearnmoreaboutSystemsofLinearEquationsinTwoVariablesandtheirGraphsthroughtheWEB.Youmayopenthefollowinglinks.

1. https://new.edu/resources/solv-ing-linear-systems-by-graphing

2. http:/ /www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/index.php

3. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-228s.html

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QU

ESTIONS?

a. Howwereyouabletoidentifysystemsofequationsthatareconsistent-dependent,consistent-independent,andinconsistent?

b. Whendoyousaythatasystemoflinearequationsisconsistentanddependent?consistentandindependent?inconsistent?

c. Giveexamplesofsystemsoflinearequationsthatareconsistentanddependent,consistentandindependent,andinconsistent.

Were you able to determine which systems of linear equations in two variables are consistent and dependent, consistent and independent, or inconsistent? In the next activity, you will describe the solution set of system of linear equations in two variables through its graph.

HOW DO I LOOK? Activity 4

Directions: Determinethesolutionsetofthesystemoflinearequationsasshownbythefollowinggraphs.Thenanswerthequestionsthatfollow.

1. 3.

2. 4.

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QU

ESTIONS?

a. Howmanysolution/sdoeseachgraphofsystemoflinearequationshave?

b. Whichgraphshowsthatthesystemoflinearequationsisconsistentanddependent?consistentandindependent?inconsistent?Explainyouranswer.

c. Whendoyousaythatthesystemoflinearequationsasdescribedbythegraphisconsistentanddependent?consistentandindependent?inconsistent?

d. Drawgraphsofsystemsoflinearequationsthatareconsistentanddependent,consistentandindependent,andinconsistent.Describeeachgraph.

Was it easy for you to describe the solution set of a system of linear equations given the graph? In the next activity, you will graph systems of linear equations then describe their solution sets.

DESCRIBE MY SOLUTIONS!Activity 5

Directions: GrapheachofthefollowingsystemsoflinearequationsintwovariablesontheCartesiancoordinateplane.Describe thesolutionsetofeachsystembasedonthegraphdrawn.Thenanswerthequestionsthatfollow.

1. x + y = 8x + y = -3

2. 3x – y = 7x + 3y = -4

3. x+6y = 92x+6y = 18

4. x – 2y = 126x + 3y = -9

5. 3x + y = -2x + 2y = -4

261

QU

ESTIONS?

a. Howdidyougrapheachsystemoflinearequationsintwovariables?b. Howdoesthegraphofeachsystemlooklike?c. Whichsystemoflinearequationshasonlyonesolution?Why? Howabout thesystemof linearequationswithnosolution? infinite

numberofsolutions?Explainyouranswer.

In this section, the discussion was about system of linear equations in two variables and their graphs.

Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Which ideas are different and need revision?

Nowthatyouknowtheimportantideasaboutthistopic,let’sgodeeperbymovingontothenextsection.

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What I have learned so far...________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________.

REFLECTIONREFLECTION

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What to UnderstandWhat to Understand Yourgoal inthissectionistotakeacloser lookatsomeaspectsofthetopic.You are going to think deeper and test further your understanding of systems oflinearequationsintwovariablesandtheirgraphs.Afterdoingthefollowingactivities,youshouldbeabletoanswerthefollowingquestion:"How is the system of linear equations in two variables used in solving real-life problems and in making decisions?"

HOW WELL I UNDERSTOOD…Activity 6

Directions: Answerthefollowing.

1. Howdoyoudescribeasystemoflinearequationsintwovariables?

2. Give at least two examples of systems of linear equations in twovariables.

3. Whenisasystemoflinearequationsintwovariablesused?

4. Howdoyougraphsystemsoflinearequationsintwovariables?

5. Howdoyoudescribethegraphsofsystemsoflinearequationsintwovariables?

6. Howdoyoudescribesystemsoflinearequationsthatareconsistentanddependent?consistentandindependent?inconsistent?

7. Studythesituationbelow:Jose wanted to construct a rectangular garden such that its

perimeteris28manditslengthis6timesitswidth.

a. What system of linearequations represents thegivensituation?

b. Supposethesystemoflinearequations is graphed. Howwouldthegraphslooklike?

c. Isthesystemconsistentanddependent, consistent andindependent,orinconsistent?Why?

264

In this section, the discussion was about your understanding of systems of linear equations in two variables and their graphs.

What new realizations do you have about the systems of linear equations in two variables and their graphs? What new connections have you made for yourself?

Now thatyouhaveadeeperunderstandingof the topic,youare ready todo thetasksinthenextsection.

What to TransferWhat to Transfer

Yourgoalinthissectionistoapplyyourlearningtoreal-lifesituations.Youwillbegivenapracticaltaskwhichwilldemonstrateyourunderstanding.

HOW MUCH AND WHAT’S THE COST?Activity 7

Directions: Complete the tablebelowbywritingall theschoolsupplies thatyouuse.Indicatethequantityandthecostofeach.

School Supply Quantity Cost

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Formulatelinearequationsintwovariablesbasedfromthetable.Thenusesomepairsoftheseequationstoformdifferentsystemsofequations.Drawthegraphofeachsystemoflinearequations.Usetherubricprovidedtorateyourwork.

Rubric for Real-Life Situations Involving Systems of Linear Equations in Two Variables and their Graphs

4 3 2 1Systematicallylistedthedatainthetable,properlyformulatedlinearequationsintwovariablesthatformasystemofequations,andaccuratelydrewthegraphofeachsystemoflinearequations.

Systematicallylistedthedatainthetable,properlyformulatedlinearequationsintwovariablesthatformasystemofequationsbutunabletodrawthegraphaccurately.

Systematicallylistedthedatainthetableandformulatedlinearequationsintwovariablesbutunabletoformsystemsofequations.

Systematicallylistedthedatainthetable.

In this section, your task was to cite three real-life situations where systems of linear equations in two variables are illustrated.

How did you find the performance task? How did the task help you see the real-world use of the topic?

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In this lesson, I have understood that ______________

________________________________________________

________________________________________________

_________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

_________________________________________________

________________________________________________

_________________________________________________

________________________________________________

___________.


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SUMMARY/SYNTHESIS/GENERALIZATION:

Thislessonwasaboutsystemsoflinearequationsintwovariablesandtheirgraphs.Thelessonprovidedyouopportunitiestodescribesystemsoflinearequationsandtheirsolutionsetsusingpracticalsituations,mathematicalexpressions,andtheirgraphs.Youidentifiedanddescribedsystemsoflinearequationswhosegraphsareparallel,intersecting,orcoinciding.Moreover,youweregiventhechancetodrawanddescribethegraphsofsystemsoflinearequationsintwovariablesandtodemonstrateyourunderstandingofthelessonbydoingapracticaltask.Yourunderstandingofthis lessonandotherpreviouslylearnedmathematicsconceptsandprincipleswill facilitateyour learningof thenext lesson,SolvingSystemsofLinearEquationsGraphicallyandAlgebraically.

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22Solving Systems of Linear Equations in

Two VariablesLesson


Start the lesson by assessing your knowledge of the different mathematicsconceptspreviouslystudiedandyourskillsinperformingmathematicaloperations.TheseknowledgeandskillsmayhelpyouinunderstandingSolvingSystemsofLinearEquationsinTwoVariables.Asyougothroughthislesson,thinkofthefollowingimportantquestion:How is the system of linear equations in two variables used in solving real-life problems and in making decisions?Tofindouttheanswer,performeachactivity.Ifyoufindanydifficultyinansweringtheexercises,seektheassistanceofyourteacherorpeersorrefertothemodulesyouhavestudiedearlier.

HOW MUCH IS THE FARE?Activity 1

Directions: Usethesituationbelowtoanswerthequestionsthatfollow.

Supposeforagivendistance,atricycledriverchargeseverypassengerPhp10.00whileajeepneydriverchargesPhp12.00.

1. Complete the table below for the fare collected by the tricycle andjeepneydriversfromacertainnumberofpassengers.

Number of Passengers

Amount Collected by the Tricycle

Driver

Amount Collected by the

Jeepney Driver1234510

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15202530

2. How did you determine the amount collected by the tricycle and

jeepneydriversfromtheirpassengers?

3. Supposein3roundtripsthetricycleandjeepneydrivershadatotalof68passengers.

a. Howwouldyoufindthenumberofpassengerseachhad?b. Whatmathematicalstatementwillyouusetofindthenumberof

passengerseachhad?c. Whatisthetotalamountoffarecollectedfromthepassengers

bythetwodrivers?Explainhowyouarrivedatyouranswer.d. Howwouldyoudrawthegraphofthemathematicalstatement

obtainedin3b?Drawanddescribethegraph.

4. SupposethetotalfarecollectedbythetricycleandjeepneydriversisPhp780.

a. Howwouldyoufindthenumberofpassengerseachcarried?b. Whatmathematicalstatementwillyouusetofindthenumberof

passengerseachhad?c. Howwouldyoudrawthegraphofthemathematicalstatement

obtained in 4b? Draw the graph in the Cartesian coordinateplanewherethegraphofthemathematicalstatementin3bwasdrawn.Describethegraph.

5. Howdoyoudescribethetwographsdrawn?

6. Whatdothegraphstellyou?

7. Howdidyoudeterminethenumberofpassengerseachdriverhad?

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How did you find the activity? Were you able to use linear equations in two variables to represent a real-life situation? Were you able to find some possible solutions of a linear equation in two variables and draw its graph? In the next activity, you will show the graphs of systems of linear equations in two variables. You need this skill to learn about the graphical solutions of systems of linear equations in two variables.

LINES, LINES, LINES…Activity 2

Directions: Drawthegraphofeachequationinthesysteminonecoordinateplane. 1.

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

3x + 8y = 128x – 5y = 12

2.

y = 3x – 28x + 7y = 15 4.

x – y = 62x + 7y=-6

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QU

ESTIONS?

a. Howdidyoushowthegraphofeachsystemofequations?b. Howdoyoudescribethegraphofeachsystemofequations?c. Arethegraphsintersectinglines?Ifyes,whatarethecoordinatesof

thepointofintersectionoftheselines?d. Whatdoyouthinkdothecoordinatesofthepointofintersectionofthe

linesmean?

Were you able to draw the graph of each system of linear equations in two variables? Were you able to determine and give the meaning of the coordinates of the point of intersection of intersecting lines? As you go through this module, you will learn about this point of intersection of two lines and how the coordinates of this point are determined algebraically. In the next activity, you will solve for the indicated variable in terms of the other variable. You need this skill to learn about solving systems of linear equations in two variables using the substitution method.

IF I WERE YOU… Activity 3

Directions: Solvefortheindicatedvariableintermsoftheothervariable.Explainhowyouarrivedatyouranswer.

1. 4x + y=11; y= 6. -2x + 7y=18; x =

2. 5x – y=9; y= 7. -3x – 8y=15; x =

3. 4x + y=12; x= 8. 14 x + 3y=2; x =

4. -5x – 4y=16; y= 9. 49 x – 1

3 y=7; y =

5. 2x + 3y=6; y= 10. - 23 x – 1

2 ;y = 8 x =

How did you find the activity? Were you able to solve for the indicated variable in terms of the other variable? In the next activity, you will solve linear equations. You need this skill to learn about solving systems of linear equations in two variables algebraically.

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Were you able to solve each equation? In solving each equation, were you able to apply the mathematics concepts or principles which you already learned? Solving equations is an important skill that you need to fully develop so you would not find difficulty in solving systems of linear equations in two variables algebraically. But how are systems of linear equations in two variables used in solving real-life problems and in making decisions? You will find these out in the activities in the next section. Before you start performing these activities, read and understand first some important notes on solving systems of linear equations and the examples presented.

WHAT MAKES IT TRUE?Activity 4

Directions: Findthevalueofthevariablethatwouldmaketheequationtrue.Answerthequestionsthatfollow.

1. 5x=15 6. x+7=10 2. -3x=21 7. 3y – 5 = 4 3. 9x=-27 8. 2y + 5y = -28

4. -7x=-12 9. -3y + 7y = 12 5. 2

3 x=8 10. 5x – 2x = -15

QU

ESTIONS?

a. Howdidyousolveeachequation?b. Whatmathematicsconceptsorprinciplesdidyouapplytosolveeach

equation?Explainhowyouappliedthesemathematicsconceptsandprinciples.

c. Doyouthinkthereareotherwaysofsolvingeachequation?Explainyouranswer.

Thesolutionofasystemoflinearequationscanbedeterminedalgebraicallyorgraphically.Tofindthesolutiongraphically,graphbothequationsinaCartesiancoordinateplanethenfindthepointof intersectionof thegraphs, if itexists.YoumayalsousegraphingcalculatororcomputersoftwaresuchasGeoGebraindeterminingthegraphicalsolutionsofsystemsoflinearequations.GeoGebraisanopen-sourcedynamicmathematicssoftwarewhichhelpsyouvisualizeandunderstandconceptsinalgebra,geometry,calculus,andstatistics.

Thesolutiontoasystemoflinearequationscorrespondstothecoordinatesofthepointsofintersectionofthegraphsoftheequations.

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Examples: Findthesolutionsofthefollowingsystemsoflinearequationsgraphically.

a. 2x + y = 7-x + y = 1 b. 3x + y = 4

3x – y = -5 c. x – 2 = -52x – 4y =-10

Answer (a): Thegraphsof2x + y = 7 and -x + y=1intersectat(2,3).

Hence, the solution of the

system 2x + y = 7-x + y = 1 is x = 2

and y=3.

Answer (b): Thegraphsof3x + y = 4 and 3x

+ y = 10 are parallel. Hence,

thesystem3x + y = 43x – y = -5 has no

solution.

Answer (c): Thegraphsofx – 2y = -5 and

2x – 4y=-10coincide.Hence,

thesystemx – 2 = -52x – 4y =-10 has

infinitenumberofsolutions.

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Asystemoflinearequationscanbesolvedalgebraicallybysubstitution or elimination methods.

Tosolveasystemoflinearequationsbysubstitutionmethod,thefollowingprocedurescouldbefollowed:

a. Solveforonevariableintermsoftheothervariableinoneoftheequations. Ifoneoftheequationsalreadygivesthevalueofonevariable,youmayproceed

tothenextstep.b. Substitutethevalueofthevariablefoundinthefirststemthesecondequation.

Simplifythensolvetheresultingequation.c. Substitute thevalueobtained in (b) toanyof theoriginalequations tofind the

valueoftheothervariable.d. Check the valuesof the variables obtainedagainst the linear equations in the

system.

Example: Solvethesystem2x + y = 5-x + 2y = 5 bysubstitutionmethod.

Solution: Use 2x + y=5tosolveforyintermsofx. Subtract-2xfrombothsidesoftheequation. 2x + y – 2x = 5 – 2x y = 5 – 2x

Substitute5–2xintheequation-x + 2y=5. -x+2(5–2x)=5

Simplify. -x+2(5)+2(-2x)=5 -x+10–4x = 5 -5x=5–10 -5x = -5

Solve for x by dividingbothsidesoftheequationby-5.

-5x-5 = -5

-5 x = 1

Substitute1,valueofx,toanyoftheoriginalequationstosolvefory. -x + 2y = 5 -1 + 2y = 5

Simplify. -1 + 2y = 5 2y = 5 + 1 2y=6

Solveforybydividingbothsidesoftheequationby2.

2y2 = 62 y = 3

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Checkthevaluesofthevariablesobtainedagainstthelinearequationsinthesystem.

1. 2x + y=5;x = 1 and y = 3 2(1)+3=2+3=5 If x=1 and y=3, theequation2x + y=5istrue. Hence,thecooordinate(1,3)satisfiestheequation. 2. -x + 2y=5;x = 1 and y = 3 -1+2(3)=-1+6=5 Ifx=1andy=3,theequation-x + 2y=5istrue. Hence,thecoordinate(1,3)satisfiestheequation.

Therefore,thesolutiontothesystem2x + y = 5-x + 2y = 5 istheorderedpair(1,3).

Tosolveasystemoflinearequationsintwovariablesbytheeliminationmethod,thefollowingprocedurescouldbefollowed:

a. Whenevernecessary,rewritebothequationsinstandardformAx + By = C.b. Whenever necessary,multiply either equation or both equations by a nonzero

numbersothatthecoefficientsof x or ywillhaveasumof0.(Note:Thecoefficientsof x and yareadditiveinverses.)

c. Addtheresultingequations.This leadstoanequationinonevariable.Simplifythensolvetheresultingequation.

d. Substitutethevalueobtainedtoanyoftheoriginalequationstofindthevalueoftheothervariable.

e. Check the valuesof the variables obtainedagainst the linear equations in thesystem.

Example: Solvethesystem3x + y = 72x – 5y =16 byeliminationmethod.

Solution: Thinkofeliminatingyfirst. Multiply5tobothsidesoftheequation3x + y=7. 5(3x + y=7) 15x + 5y = 35

Addtheresultingequations. 15x + 5y = 35

2x – 5y =1617x = 51

Solve for xbydividingbothsidesoftheequationby17.

17x = 51 17x17 = 51

17 x = 3 Substitute3,valueofx,toanyoftheoriginalequationstosolvefory. 2x – 5y=16 2(3)–5y=16 Simplify. 6–5y=16 -5y=16–6 -5y =10

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Solve for ybydividingbothsidesoftheequationby-5.

-5y=10 -5y-5 = 10-5 y = -2

Checkthevaluesofthevariablesobtainedagainstthelinearequationsinthesystem.

1. 3x + y=7;x = 3 and y = -2 3(3)+(-2)=9–2=7 If x = 3 and y =-2,theequation3x + y=7istrue. Hencethecoordinate(3-2)satisfiestheequation. 2. 2x – 5y=16;x = 3 and y = -2 2(3)–5(-2)=6+10=16 If x = 3 and y =-2,theequation2x – 5y=16istrue. Hence,thecoordinate(3,-2)satisfiestheequation. Therefore,thesolutiontothesystem

3x + y = 72x – 5y =16 istheorderedpair(3,-2).

Systemsof linearequations in twovariablesareapplied inmany real-life situations.Theyareusedtorepresentsituationsandsolveproblemsrelatedtouniformmotion,mixture,investment,work,andmanyothers.Considerthesituationbelow.

A computer shop hires 12 technicians and 3 supervisors for total daily wages of Php 7,020. If one of the technicians is promoted to a supervisor, the total daily wages become Php 7,110.

Inthegivensituation,whatdoyouthinkisthedailywageforeachtechnicianandsuper-visor?Thisproblemcanbesolvedusingsystemoflinearequations.

Letx=dailywageofatechnicianandy=dailywageofasupervisor.Representthetotaldailywagesbeforeoneofthetechniciansispromotedtoasupervisor.

12x + 3y=7,020

Representthetotaldailywagesafteroneofthetechniciansispromotedtoasupervisor. 11x + 4y=7,110

Usethetwoequationstofindthedailywagesforatechnicianandasupervisor.

12x + 3y =7,020 11x + 4y =7,110

Solvethesystemgraphicallyorbyusinganyalgebraicmethod.

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Let’ssolvethesystemusingEliminationMethod.Multiplybothsidesofthefirstequationby4andthesecondequationby3toeliminatey.

12x + 3y = 7,02011x + 4y =7,110 4(12x + 3y = 7,020)

3(11x + 4y =7,110) 48x + 12y = 28,08033x + 12y =21,330

Theresultingsystemoflinearequationsis48x + 12y = 28,08033x + 12y =21,330

. Subtractthetermsonbothsidesoftheresultingequations.

48x + 12y = 28,08033x + 12y =21,33015x =6,750

Usingtheequation15x=6,750,solveforxbydividingbothsidesoftheequationby15.

15x =6,750 15x15 = 6,75015 x=450

ThedailywageofatechnicianisPhp 450.

Findthedailywageofasupervisorbysubstituting450forx in anyoftheoriginalequations.Then,solvetheresultingequation.

12x + 3y=7,020; x=450 12(450)+3y=7,020 5,400+3y=7,020 3y=7,020–5,400 3y=1,620

3y3 = 1,6203 y=540

ThedailywageofasupervisorisPhp 540.

Answer: ThedailywagesforatechnicianandasupervisorarePhp 450 and Php 540,respectively.

Youhaveseenhowasystemoflinearequationsisusedtosolveareal-lifeproblem.Inwhatotherreal-lifesituationsaresystemsoflinearequationsintwovariablesillustratedorapplied?Howisthesystemoflinearequationsintwovariablesusedinsolvingreal-lifeproblemsandinmakingdecisions?


1. http://www.mathguide.com/les-sons/Systems.html

2. http:/ /www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/index.php

3. http://edhelper.com/LinearEqua-tions.htm

4. http://www.purplemath.com/modules/systlin1.htm






10.h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-236s.html

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Now that you learned about solving systems of linear equations in two variables graphically and algebraically, you may now try the activities in the next section.


Your goal in this section is to learn and understand solving systems of linearequationsgraphicallyandalgebraically.Usethemathematicalideasandtheexamplespresentedintheprecedingsectioninansweringtheactivitiesprovided.

WHAT SATISFIES BOTH?Activity 5

Directions: Solve each of the following systems of linear equations graphically, thencheck.YoumayalsouseGeoGebratoverifyyouranswer.Ifthesystemoflinearequationshasnosolution,explainwhy.

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

2y = 4 –6x

2. x – y = 5x + 5y = -7 4. x + y = 4

2x – 3y = 3

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5. y = 5x – 25x – 3y = -14 6. 2x – 3y = 5

3y=10 + 2x

Were you able to determine the solution of each system of linear equations in two variables graphically? In the next activity, you will determine the resulting equation when the value of one variable is substituted to a given equation.

How did you find the activity? Do you think it would help you perform the next activity? Find out when you solve systems of linear equations using the substitution method.

TAKE MY PLACE!Activity 6

Directions: Determine the resulting equation by substituting the given value of onevariabletoeachofthefollowingequations.Thensolvefortheothervariableusingtheresultingequation.Answerthequestionsthatfollow.

Equation Value of Variable Equation Value of Variable1.4x + y=7;

2.x + 3y=12;

3.2x – 3y=9;

y = x + 3

x = 4 – y

y = x – 2

4.5x + 2y=8; 5.4x – 7y=-10;

6.-5x = y–4;

x = 3y + 1

y = x – 4

y = 3x + 5

QU

ESTIONS?

a. Howdidyoudetermineeachresultingequation?b. Whatresultingequationsdidyouarriveat?c. Howdidyousolveeachresultingequation?d. Whatmathematicsconceptsorprinciplesdidyouapplytosolveeach

resultingequation?e. Howwillyoucheckifthevalueyougotisasolutionoftheequation?

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ELIMINATE ME!Activity 8

Directions: Determinethenumber(s)thatmustbemultipliedtooneorbothequationsin each system to eliminate one of the variables by adding the resultingequations.Justifyyouranswer.

1. x – y = -33x + y = 19 4. x + 3y = 5

4x + 2y = 7 2. 2x + y = 7

-2x + 3y = 5

5. 2

3 x + 5y = 10

3x – 54y = 1 3. 5x – 2y = 122x + y = 7

Were you able to find the solution set of each system of linear equations? Do you think this is the most convenient way to solve a system of equations? In the next activity, you will determine the number(s) that must be multiplied to the terms of one or both equations in a system of equations. This will lead you finding the solution set of a system of linear equations in two variables using the elimination method.

SUBSTITUTE THEN SOLVE!Activity 7

Directions: Determine the resultingequation ifonevariable issolved in termsof theother variable in one equation, and substitute this variable in the otherequation.Thensolvethesystem,andanswerthequestionsthatfollow.

1. x + y = 8y = x +6 6. 3x + y = 2

9x + 2y = 7

2. x = -y + 7x – y = -9 7. x – y = -3

3x + y = 19

3. y = 2x4x + 3y =20 8. 4x + y = 6

x – 2y = 15

4. y = 2x + 53x – 2y = -5 9. 2x + y =10

4x + 2y = 5

5. 2x + 5y = 9-x + y = 2 10. -x + 3y = -2

-3x + 9y =-6

QU

ESTIONS?

a. Howdid you use substitutionmethod in finding the solution set ofeachsystemoflinearequations?

b. Howdidyoucheckthesolutionsetyougot?c. Whichsystemsofequationsaredifficulttosolve?Why?d. Whichsystemsofequationshavenosolution?Why?e. Which systems of equations have an infinite number of solutions?

Explainyouranswer.

To eliminate x

To eliminate y

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6. -3x + 2y = 75x + 2 = 4y 9. 2x + 3y = 6

4x+6y = 12

7. 9x – 5y = 87y + 3x = 12 10. 14x–6y – 5 = 0

6x+10y –1=0

8. 12x + 5y = 215x – 15y = 1

How did you find the activity? Do you think it will help you perform the next activity? Find out when you solve systems of linear equations using the elimination method.

ELIMINATE THEN SOLVE!Activity 9

Directions: Solveeachsystemoflinearequationsbytheeliminationmethod,thencheckyouranswers.Answerthequestionsthatfollow.

1. 3x + 2y = -42x – y = -12 6. 3x + 7y = 12

5x – 4y =20

2. 7x – 2y = 45x + y = 15 7. 2x + y = 9

x – 2y =6

3. 5x + 2y =6-2x + y =-6 8. 5x + 2y = 10

3x – 7y = -4

4. 2x + 3y = 73x – 5y = 1 9. 2x + 7y = -5

3x – 8y = -5

5. x – 4y = 93x – 2y = 7 10. -3x + 4y = -12

2x – 5y =6

QU

ESTIONS?

a. Howdidyouusetheeliminationmethodinsolvingeachsystemoflinearequations?

b. Howdidyoucheckyoursolutionset?c. Whichsystemsofequationsweremostdifficulttosolve?Why?d. Whenistheeliminationmethodconvenienttouse?e. Amongthethreemethodsofsolvingsystemsoflinearequationsintwo

variables,whichdoyouthinkisthemostconvenienttouse?Whichdoyouthinkisnot?Explainyouranswer.

In this section, the discussion was about solving systems of linear equations in two variables by using graphical and algebraic methods. Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Which ideas are different and need revision? Now that you know the important ideas about solving systems of linear equations in two variables, let’s go deeper by moving on to the next section.

To eliminate x

To eliminate y

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What I have learned so far...________________________

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What to UnderstandWhat to Understand Yourgoalinthissectionistotakeacloserlookatsomeaspectsofthetopic.Youaregoingtothinkdeeperandtestfurtheryourunderstandingofthedifferentmethodsofsolvingsystemsoflinearequationsintwovariables.Afterdoingthefollowingactivities,youshouldbeable toanswer the followingquestion:How is the system of linear equations in two variables used in solving real-life problems and in making decisions?

LOOKING CAREFULLY AT THE GRAPHS…Activity 10

Directions: Answerthefollowingquestions:

1. Howdoyoudeterminethesolutionsetofasystemoflinearequationsfromitsgraph?

2. Doyou think it iseasy todetermine thesolutionsetofasystemoflinearequationsbygraphing?Explainyouranswer.

3. When are the graphical solutions of systems of linear equationsdifficulttodetermine?

4. Howwouldyoucheckifthesolutionsetyoufoundfromthegraphsofasystemoflinearequationsiscorrect?

5. Whatdoyouthinkaretheadvantagesandthedisadvantagesofthegraphicalmethodofsolvingsystemsoflinearequations?Explainyouranswer.

Were you able to answer all the questions in the activity? Do you have better understanding of the graphical method of solving systems of linear equations? In the next activity, you will be given the opportunity to deepen your understanding of solving systems of linear equations using the substitution method.

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HOW SUBSTITUTION WORKS…Activity 11

Directions: Usethesystemoflinearequations5x – 2y = 32x + y = 12 toanswerthefollowing:

1. Howwouldyoudescribeeachequationinthesystem?2. Howwillyousolvethegivensystemofequations?3. Doyouthinkthatthesubstitutionmethodismoreconvenienttousein

findingthesolutionsetofthesystem?Explainyouranswer.4. Whatisthesolutionsetofthegivensystemofequations?Explainhow

youarrivedatyouranswer.5. Whenisthesubstitutionmethodinsolvingsystemsoflinearequations

convenienttouse?6. Givetwoexamplesofsystemsoflinearequationsintwovariablesthat

areeasytosolvebysubstitution?Solveeachsystem.

How did you find the activity? Were you able to have a better understanding of the substitution method of solving systems of linear equations? In the next activity, you will be given the opportunity to deepen your understanding of solving systems of linear equations using the elimination method.

The activity provided you with opportunities to deepen your understanding of solving systems of linear equations in two variables using the elimination method. You were able to find out which systems of linear equations can be solved conveniently by using the substitution or elimination method. In the next activity, you will extend your understanding of systems of linear equations in two variables to how they are used in solving real-life problems.

ELIMINATE ONE TO FIND THE OTHER ONE Activity 12

Directions: Use the system of linear equations 3x – 5y = 82x + 7y =6 to answer the following

questions:

1. Howwouldyoudescribeeachequationinthesystem?2. Howwillyousolvethegivensystemofequations?3. Whichalgebraicmethodofsolvingsystemoflinearequationsdoyou

thinkismoreconvenienttouseinfindingitssolutionset?Why?4. Whatisthesolutionsetofthegivensystemofequations?Explainhow

youarrivedatyouranswer.5. Whenistheeliminationmethodinsolvingsystemsoflinearequations

convenienttouse?6. Givetwoexamplesofsystemsoflinearequationsintwovariablesthat

areeasytosolvebyelimination.Solveeachsystem.

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SOLVE THEN DECIDE!Activity 13

Directions: Answereachofthefollowingquestions.Showyourcompletesolutionsandexplanations/justifications.

1. Whichof the following ismoreeconomicalwhen rentingavehicle?Justifyyouranswer.

LG’sRentaCar: Php1,500perdayplusPhp35perkilometertraveled RentandDrive: Php2,000perdayplusPhp25perkilometertraveled

2. LuisasellstwobrandsoftabletPCs.Shereceivesacommissionof12%onsalesforBrandAand8%onsalesforBrandB.IfsheisabletoselloneofeachbrandoftabletforatotalofPhp42,000,shewillreceiveacommissionofPhp4,400.a. WhatisthecostofeachbrandoftabletPC?b. Howmuchcommissiondidshe receive from thesaleofeach

brandoftablet?c. SupposeyouareLuisaandyouwanttomaximizeyourearnings.

WhichbrandoftabletPCwillyouselltomaximizeyourearnings.WhichbrandoftabletPCwillyouencourageyourclientstobuy?Why?

3. CaraandTrishaarecomparingtheirplansforWorldCelcompostpaid

subscribers.ShouldCaraswitchtoTrisha'splan?Justifyyouranswer. Cara'splan: Php500monthlycharge FreecallsandtextstoWorldCelcomsubscribers Php6.50perminuteforcallstoothernetworks Trisha'splan: Php650monthlycharge FreecallsandtextstoWorldCelcomsubscribers Php5.00perminuteforcallstoothernetworks

4. Mr.Salongahastwoinvestments.HistotalinvestmentisPhp400,000.Annually, he receives 3% interest on one investment and 7%interestontheother.ThetotalinterestthatMr.SalongareceivesinayearisPhp16,000.

a. HowmuchmoneydoesMr.Salongahaveineachinvestment?b. InwhichinvestmentdidMr.Salongaearnmore?c. Suppose youwereMr.Salonga, inwhich investmentwill you

placemoremoney?Why?

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What new insights do you have about solving systems of linear equations? What new connections have you made for yourself?

Let’s extend your understanding. This time, apply what you have learned in real life by doing the tasks in the next section.


Yourgoal in this section is toapply your learning to real-life situations.Youwillbegivenapractical task inwhichyouwilldemonstrateyourunderstandingofsolvingsystemsoflinearequationsintwovariables.

PLAY THE ROLE OF …Activity 14

Citesituationsinreallifewheresystemsoflinearequationsintwovariablesareapplied.Formagroupof5membersandroleplayeachsituation.Withyourgroupmates,formulateproblemsoutofthesesituations,thensolvetheminasmanywaysasyoucan.

SELECT THE BEST POSTPAID PLANActivity 15

1. Makealistofallpostpaidplansbeingofferedbydifferentmobilenetworkcompanies.

2. Usethepostpaidplanstoformulateproblemsinvolvingsystemsoflinearequationsintwovariables.Clearlydefineallvariablesusedandsolveallproblemsformulated.Usethegivenrubrictorateyourwork.

3. Determine the best postpaid plan that each company offers based on your currentcellphoneusage.Explainyouranswer.

5. Theschoolcanteensellschickenandeggsandwiches.Itgeneratesa revenueofPhp2 foreverychickensandwichsoldandPhp1.25for every egg sandwich sold. Yesterday, the canteen sold all 420sandwiches that the staff prepared and generated a revenue ofPhp615.

a. Howmanysandwichesofeachkindwasthecanteenabletosellyesterday?

b. Supposetheteacherinchargeofthecanteenwishestoincreasethecanteen's revenue fromsandwichessold toPhp720. Is itpossibletodothiswithoutraisingthepricepersandwich?How?

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In this section, your tasks were to cite real-life situations and formulate and solve problems involving systems of linear equations in two variables.

How did you find the performance task? How did the task help you see the real world application of systems of linear equations in two variables?

Rubric on Problems Formulated and Solved

Score Descriptors6 Poses amore complex problemwith 2 ormore correct possible solutions

and communicates ideas unmistakably, shows in-depth comprehension ofthepertinentconceptsand/orprocessesandprovidesexplanationswhereverappropriate.

5 Posesamorecomplexproblemandfinishesallsignificantpartsofthesolu-tionandcommunicatesideasunmistakably,showsin-depthcomprehensionofthepertinentconceptsand/orprocesses.

4 Poses a complex problem and finishes all significant parts of the solutionandcommunicatesideasunmistakably,showsin-depthcomprehensionofthepertinentconceptsand/orprocesses.

3 Posesacomplexproblemandfinishesmostsignificantpartsofthesolutionandcommunicatesideasunmistakably,showscomprehensionofmajorcon-ceptsalthoughneglectsormisinterpretslesssignificantideasordetails.

2 Posesaproblemandfinishessomesignificantpartsofthesolutionandcom-municatesideasunmistakablybutshowsgapsontheoreticalcomprehension.

1 Posesaproblembutdemonstratesminorcomprehension,notbeingabletodevelopanapproach.

Source:D.O.#73s.2012

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This lessonwasaboutsolvingsystemsof linearequationsintwovariablesusingthegraphicalandalgebraicmethodsnamely:substitutionandeliminationmethods.Inthislesson,youwereabletofinddifferentwaysoffindingthesolutionsofsystemsoflinearequationsandgiventheopportunitytodeterminetheadvantagesanddisadvantagesofusingeachmethodandwhichismoreconvenienttouse.Usingthedifferentmethodsofsolvingsystemsoflinearequations,youwereabletofindoutwhichsystemhasnosolution,onesolution,andinfinitenumberofsolutions.Moreimportantly,youweregiventhechancetoformulateandsolvereal-lifeproblems,makedecisionsbasedontheproblems,anddemonstrateyourunderstandingofthelessonbydoingsomepracticaltasks.Yourunderstandingofthislessonwillbeextend-edinthenextlesson,GraphicalSolutionsofSystemsofLinearInequalitiesinTwoVariables.Themathematicalskillsyouacquiredinfindingthegraphicalsolutionsofsystemsof linearequationscanalsobeappliedinthenextlesson.

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33Graphical Solutions

of Systems of Linear Inequalities in Two

Variables

Lesson


Start Lesson 3 of this module by assessing your knowledge of the differentmathematics concepts previously studied and your skills in performing mathematicaloperations.TheseknowledgeandskillsmayhelpyouinunderstandingGraphicalSolutionsofSystemsofLinearInequalitiesinTwoVariables.Asyougothroughthislesson,thinkof the following importantquestion:How is the system of linear inequalities in two variables used in solving real-life problems and in making decisions?Tofindouttheanswer,performeachactivity. Ifyoufindanydifficulty inanswering theexercises,seektheassistanceofyourteacherorpeersorrefertothemodulesyouhavegoneoverearlier.

SUMMER JOBActivity 1

Directions: Usethesituationbelowtoanswerthequestionsthatfollow.

Nimfa livesnearabeachresort.Duringsummervacation,shesellssouveniritemssuchasbraceletsandnecklacesmadeoflocalshells.Eachbracelet costsPhp85while eachpieceof necklace costsPhp115.SheneedstosellatleastPhp15,000worthofbraceletsandnecklaces.

QU

ESTIONS?

a. Howdidyouuse theeliminationmethod insolvingeachsystemoflinearequations?

b. Howdidyoucheckthesolutionsetyougot?c. Whichsystemofequationsisdifficulttosolve?Why?d. Whenistheeliminationmethodconvenienttouse?e. Among the three methods of solving systems of linear equations

intwovariables,whichdoyouthinkisthemostconvenienttouse?Whichdoyouthinkisnot?Explainyouranswer.

QU

ESTIONS? 1. Completethetablebelow.

Number of bracelets sold

Cost Number of necklaces sold

Cost Total Cost

1 12 23 3

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4 45 510 1015 1520 2025 2530 3040 4050 5060 6080 80100 100

2. HowmuchwouldNimfa’stotalsalebeifshesells5braceletsand5necklaces?

Whataboutifshesells10braceletsand20necklaces?

3. What mathematical statement would represent the total sale ofbracelets and necklaces? Describe the mathematical statement,thengraphitfortheparticularcasewhenNimfamakesatotalsaleofPhp15,000.

4. Nimfa wants to have a total sale of at least Php 15,000. Whatmathematical statement would represent this? Describe themathematicalstatement.thengraph.

5. HowmanybraceletsandnecklacesshouldNimfaselltohaveatotalsaleofatleastPhp15,000?Giveasmanyanswersaspossiblethenjustify.

How did you find the activity? Were you able to use linear inequalities in two variables to represent a real-life situation? Were you able to find some possible solutions of a linear inequality in two variables and draw its graph? In the next activity, you will recall what you learned about graphing linear equations and inequalities. You will need this skill in finding the graphical solution of a system of linear inequalities in two variables.

292

A LINE OR HALF OF A PLANE?Activity 2

Directions: Draw thegraphsof the following linearequationsand inequalities in twovariables.Answerthequestionsthatfollow.

1. 3x + y=10 2. 5x – y = 12 3. 2x + 3y = 15 4. 3x – 4y = 8 5. 4x + 7y = -8

6. 3x + y<10 7. 5x – y > 12 8. 2x + 3y≤15 9. 3x – 4y≥8 10. 4x + 7y < -8

QU

ESTIONS?

a. Howdidyougrapheachmathematicalstatement?b. Comparethegraphsof3x + y=10and3x + y<10.Whatstatements

canyoumake? Howabout 5x – y = 12 and 5x – y > 12? 2x + 3y = 15 and 2x + 3y≤15?c. How would you differentiate the graphs of linear equations and

inequalitiesintwovariables?d. Howmanysolutionsdoesa linearequation in twovariableshave?

Howaboutlinearinequalitiesintwovariables?e. Supposeyoudrewthegraphsof3x + y<10and 5x – y>12inanother

Cartesiancoordinateplane.Howwouldyoudescribe theirgraphs?Whatorderedpairswouldsatisfybothinequalities?

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Were you able to draw the graph of each mathematical statement? Were you able to compare the graphs of linear equations and inequalities in two variables? Were you able to find ordered pairs that satisfy two linear inequalities? Finding solutions of a linear inequality leads you to understand the graphical solution of a system of linear inequalities in two variables.

To prepare yourself for the activities that follow, first, read and understand some important notes on the Graphical Solutions of System of Linear Inequalities in Two Variables. Acquaint yourself with the solutions of the examples presented so that you can answer the next activities successfully.

An ordered pair (x, y) is a solution to a system of inequalities if it satisfies all theinequalitiesinthesystem.Graphically,thecoordinatesofapointthatlieonthegraphsofallinequalitiesinthesystemispartofitssolution.

Tosolveasystemofinequalitiesintwovariablesbygraphing,

1. Draw the graph of each inequality on the same coordinate plane. Shade theappropriate half-plane. Recall that if all points on the line are included in thesolution,itisaclosedhalfplane,andthelineissolid.Ontheotherhand,ifthepointsonthelinearenotpartofthesolutionoftheinequality,itisanopenhalf-planeandthelineisbroken.

2. Theregionwhereshadedareasoverlapisthegraphicalsolutiontothesystem.Ifthegraphsdonotoverlap,thenthesystemhasnosolution.

Example: Tosolvethesystem2x – y > -3x + 4y≤9 graphically,graph2x – y > -3 and

x + 4y≤9on thesameCartesiancoordinateplane.The regionwheretheshadedregionsoverlapisthegraphofthesolutiontothesystem.

Example: There are atmost 56 people composed of children and adultswhoare riding inabus.EachchildandadultpaidPhp80andPhp100,respectively.IfthetotalamountcollectedwasnotmorethanPhp4,800,howmanychildrenandadultscantherebeinthebus?

Likesystemsof linearequationsin two variables, systems of linearinequalities may also be applied tomany real-life situations. They areusedtorepresentsituationsandsolveproblems related to uniform motion,mixture, investment, work, and manyothers.

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Solution: Letx=numberofchildreninthebus y=numberofadultsinthebus

Representthenumberofpeopleinthebusasx + y≤56. Representtheamountcollectedas80x+100y≤4,800.

Thus,thesystemassociatedwiththeexampleisactuallycomposedoffourlinearinequalities: x + y≤56

80x+100y≤4,800 x≥0 y≥0

The region where the shaded regions overlap is the graph of thesolution of the system. Consider any point in this shaded region, thensubstituteitscoordinatesinthesystemtocheck.

Considerthepointwhosecoordinatesare(20,30).Checkthisagainsttheinequalitiesx + y≤56and80x+100y≤4,800.

If x=20andy=30,then20+30≤56.Thefirstinequalityissatisfied.

If x=20andy=30,then80(20)+100(30)=4,600≤4,800. Asidefromthis,(20,30)isalsoapointthatliesontheregionwhere

theshadedareasoverlap.Thismeansthatunderthegivenconditions,itispossiblethat20childrenand30adultsareinthebus.

Arethereanyotherconditionsthatyouthinkshouldbesatisfied?Notethat (-2, -5) isalsoapoint that ison the regionwhere theshadedareasintersect.Doesthisanswermakesenseinthecontextofthegivenproblem?Youarerightifyouthinkthatitdoesnotmakesense.

Sincexwasdefinedtobethenumberofchildreninthebus,x = -2 is meaningless.Infact,anotherconstraintfortheproblemisthatx≥0sincetherecanneverbeanegativenumberofchildreninsidethebus.

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Similarly,sincey wasdefinedtobethenumberofadultsinthebus,y=-5ismeaningless.Infact,anotherconstraintfortheproblemisthaty≥0sincetherecanneverbeanegativenumberofadultsinsidethebus.


1. http://www.purplemath.com/modules/syslneq.htm

2. https://new.edu/resources/solv-ing-systems-of-linear-inequali-ties-two-variables

3. h t tp : / /www.netp laces .com/algebra-guide/graphing-linear-relationships/graphing-linear-inequalities-in-two-variables.htm



Now that you have learned about the graphical solutions of systems of linear inequalities in two variables, you may try the activities in the next section.


Yourgoalinthissectionistolearnandunderstandhowsystemsoflinearinequalitiesintwovariablesaresolvedgraphically.Usethemathematicalideasandtheexamplespresentedinansweringthesucceedingactivities.

DO I SATISFY YOU?Activity 3

Directions: Determineifeachorderedpairisasolutionofthesystemoflinearinequality

2x + 5y <103x – 4y≥-8 .Then,answerthequestionsthatfollow.

1. (3,5) 6. (2,15) 2. (-2,-10) 7. (-6,10) 3. (5,-12) 8. (-12,1) 4. (-6,-8) 9. (0,2) 5. (0,0) 10. (5,0)

QU

ESTIONS?

a. Howdidyoudetermineifthegivenorderedpairisasolutionofthesystem?

b. Howdidyouknowthatthegivenorderedpairisnotasolutionofthesystem?

c. Howmanysolutionsdoyouthinkdoesthegivensystemofinequalitieshave?

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Were you able to find out which ordered pairs are solutions of the given system of linear inequalities in two variables? In the next activity, you will determine the graphical solutions of systems of linear inequalities in two variables.

REGION IN A PLANE Activity 4

Directions: Answerthefollowingquestions.1. Showthegraphofthesolutionofthesystem

2x + 5y < 153x – y≥8 .Usethe

Cartesiancoordinateplanebelow..

2. Howwouldyoudescribethegraphsof2x + 5y < 15 and 3x – y≥8?3. Howwouldyoudescribetheregionwherethegraphsof2x + 5y < 15

and 3x – y≥8meet?4. Selectanythreepointsintheregionwherethegraphsof2x + 5y < 15 and

3x – y≥8meet.Whatstatementscanyoumakeaboutthecoordinatesofthesepoints?

5. Howwouldyoudescribethegraphicalsolutionofthesystem2x + 5y < 153x – y≥8

?

6. How is the graphical solution of a system of linear inequalitiesdetermined?

Howisitsimilarordifferentfromthegraphicalsolutionofasystemoflinearequations?

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AM I IN THAT REGION?Activity 5

Directions: Solvethefollowingsystemsofinequalitiesgraphically.Findthreepointsthatsatisfybothinequalities.Plotthepointstoshowthattheybelongtothesolutionofthesystem.Thefirstonewasdoneforyou.

1.5x + y > 3y≤x – 4 3.

2x – y ≥-2y < x + 4

2.x + y ≥73x – y≤10 4.

y > 2x – 9y < 4x + 1

Someorderedpairssatisfyingthesystemofinequalitiesare(10,2),(5,-4),and(10,-9).

Were you able to answer all the questions in the activity? Do you now have a better understanding of the graphical solution of a system of linear inequalities in two variables? In the next activity, you will be given the opportunity to deepen your understanding.

298

5.x + y < 12y < -3x + 5 7.

x + 3y > 9x – 3y≤9

6.y > 2x + 72x – y < 12 8.

2x – y≥102y≥5x + 1

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LOOKING CAREFULLY AT THE REGION…Activity 6

Directions: Answerthefollowingquestions.

1. Howdoyoudeterminethesolutionsetofasystemoflinearinequalitiesintwovariablesfromitsgraph?

2. Doyou think it iseasy todetermine thesolutionsetofasystemoflinearinequalitiesbygraphing?Explainyouranswer.

3. Inwhatinstancewillyoufinditdifficulttodeterminethesolutionsetofasystemoflinearinequalitiesfromitsgraph?

4. Howwouldyouknow if thesolutionsyou found from thegraphsoflinearinequalitiesinasystemaretrue?

QU

ESTIONS?

a. Howdidyoudeterminethegraphicalsolutionofeachsystemoflinearinequalitiesintwovariables?

b. Howdidyouknowthattheorderedpairsyoulistedaresolutionsofthesystemofinequalities?

c. Whichsystemoflinearinequalitieshasnosolution?Why?d. Whencanyousaythatasystemoflinearinequalitieshasasolution?

nosolution?e. Give2examplesofasystemof linear inequalities in twovariables

thatdonothaveanysolution.Justifyyouranswer.

9.2x – y < 113x + 5y≥8 10. 6x + 2y≥9

3x + y≤-6

300

These activities provided you with opportunities to deepen your understanding of solving systems of linear inequalities in two variables graphically. In the next activity, you will extend your understanding to find out how these systems are used in solving real-life problems and in making decisions.

5. What do you think are the advantages and the disadvantages offindingthesolutionsetofasystemoflinearinequalitiesgraphically?Explainyouranswer.

6. Isitpossibletofindthesolutionsetofasystemoflinearinequalitiesintwovariablesalgebraically?Giveexamplesifthereareany.

SOLVE THEN DECIDE!Activity 7

Directions: Answereachofthefollowing.Showyourcompletesolutionsandexplanations.

1. TicketsforaplaycostPhp250foradultsandPhp200forchildren.ThesponsoroftheshowcollectedatotalamountofnotmorethanPhp44,000frommorethan150adultsandchildrenwhowatchedtheplay.

a. Whatmathematicalstatementsrepresentthegivensituation?b. Drawanddescribethegraphsofthemathematicalstatements.c. Howcanyoufind thepossiblenumberof childrenandadults

whowatchedtheplay?d. Give4possiblenumbersofadultsandchildrenwhowatchedthe

play.Justifyyouranswers.e. Thesponsoroftheshowrealizedthatifthepricesofthetickets

werereduced,morepeoplewouldhavewatchedtheplay.Ifyouwerethesponsoroftheplay,wouldyoureducethepricesofthetickets?Why?

2. Mr.AgoncillohasatleastPhp150,000depositedintwobanks.Onebankgivesanannualinterestof4%whiletheotherbankgives6%.Inayear,Mr.AgoncilloreceivesatmostPhp12,000.

a. Whatmathematicalstatementsrepresentthegivensituation?b. Drawanddescribethegraphsofthemathematicalstatements.c. Howwillyoudeterminetheamountdepositedineachbank?d. GivefourpossibleamountsMr.Agoncillocouldhavedeposited

ineachbank.Justifyyouranswers.e. IfyouwereMr.Agoncillo,inwhatbankaccountwouldyouplace

greateramountofmoney?Why?

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3. Mrs.Burgoswants tobuyat least30kilosofporkandbeef forherrestaurantbusinessbutcanspendnomorethanPhp12,000.AkiloofporkcostsPhp180andakiloofbeefcostsPhp220.

a. Whatmathematicalstatementsrepresentthegivensituation?b. Drawanddescribethegraphsofthemathematicalstatements.c. HowwillyoudeterminetheamountofporkandbeefthatMrs.

Burgosneedstobuy?d. Give4possibleamountsofporkandbeefthatMrs.Burgoscan

buy.Justifyyouranswers.

4. RonaldneedstoearnatleastPhp2,500fromhistwojobstocoverhisweeklyexpenses.Thisweek,hecanworkforatmost42hours.HisjobasagasstationattendantpaysPhp52.50perhourwhilehisjobasparkingattendantpaysPhp40perhour.

a. Writeasystemoflinearinequalitiestomodelthegivensituation?b. Giventhisconditions,canRonaldbeabletomeethistargetof

earningPhp2,500?Whyorwhynot?Justifyyouranswer.

5. Jane isbuyingsquidballsandnoodles forher friends.EachcupofnoodlescostsPhp15whileeachstickofsquidballscostsPhp10.SheonlyhasPhp70butneedstobuyatleast3sticksofsquidballs.

a. Writeasystemoflinearinequalitiestomodelthegivensituation.b. Solvethesystemgraphicallyc. Find at least 3 possible numbers of sticks of squid balls and

cupsofnoodlesthatJanecanbuy.Justifyyouranswers.

What new insights do you have about the graphical solutions of systems of linear inequalities in two variables? What new connections have you made for yourself?

Let’s extend your understanding. This time, apply to real-life situations what you have learned by doing the tasks in the next section.

What to UnderstandWhat to Understand

Yourgoalinthissectionistotakeacloserlookatsomeaspectsofthegraphicalsolutionsofsystemsof linear inequalities in twovariables.Afterdoing the followingactivities,youshouldbeabletoanswerthefollowingquestion:“How is the system of linear inequalities in two variables used in solving real-life problems and in making decisions?”

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JOIN THE CAMP!Activity 9

YouarechosentobeoneofthemembersoftheBoyScoutsofthePhilippineswhowillrepresentyourschoolintheNationalJamboreenextmonth.Yourscoutmasterassignedyoutotakechargeofallthecampingmaterialsneededforthetrip.Thesematerialsincludetent,ropes,cookingutensils,firewood,aswellasotheritemsyoumaythinkarenecessary.Healsoaskedyoutoprepareamenuforthefirst3daysofthejamboreeandspecifytheingredientsthatyouwillneed.

1. Makealistofallcampingmaterialsneeded.Specifythequantityforeach,aswellasitsprice,ifavailable

2. Makea listofall ingredientsyouwillneed foryourchosenmenu.Specifyquantitiesneededandtheunitpriceforeachingredient.

3. Setapossibleamountthatyourscoutmasterwillgiveyoutobuyalltheingredients.4. Usethedatafrom(1),(2),and(3)toformulateatleast5problemsinvolvingsystems

oflinearinequalitiesintwovariables.Solveeachproblemandusethegivenrubrictocheckthequalityofyourwork.

Rubric on Problems Formulated and Solved

Score Descriptors6 Posesatleast5problemswithcompleteandaccuratesolutions.

Definesallvariablesusedclearlyandaccurately.Communicatesideasclearly,showsin-depthcomprehensionofthepertinentconceptsand/orprocessesinthislesson.Providesexplanationswheneverappropriate.


Yourgoalinthissectionistoapplyyourlearningtoreal-lifesituations.Youwillbegivenapracticaltaskwhichwilldemonstrateyourunderstandingofthegraphicalsolu-tionsofsystemsoflinearinequalitiesintwovariables.

PLAY THE ROLE OF …Activity 8

Formagroupof3membersandthinkofatleast3real-lifesituationswheresystemsoflinearinequalitiesintwovariablescanbeapplied.Formulateproblemsoutofthesesituationsandsolveeach.Presentyourfindingstotheclass.

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How did you find the performance task? Did the task help you see the real-world applications of systems of inequalities in two variables? What important things have you learned from the activity? What values can be practiced through this task?

5 Posesatleast5problems,atleast4ofwhichhavecompleteandaccuratesolutions.Definesallvariablesusedclearlyandaccurately.Communicatesideasclearly,showsin-depthcomprehensionofthepertinentconceptsand/orprocessesinthislesson.

4 Posesatleat4problems,atleast3ofwhichhavecompleteandaccuratesolutionsOR commitsnomorethan3minorerrors(e.g.,wrongsign, lackofproperunits,etc.)Definesmostvariablesusedclearlyandaccurately.Communicates most ideas clearly, shows in-depth comprehension of thepertinentconceptsand/orprocessesinthislesson.

3 Posesatleast3problems,atleast2ofwhichhavecompletesolutionsANDcommitsnomorethan4minorerrors(e.g.,wrongsign,lackofproperunits,etc.)Definesmostvariablesusedclearlyandaccurately.Communicatesideasclearly,andshowscomprehensionofthemajorconceptsand/orprocessesinthislesson,butneglectsormisinterpretslesssignificantideasordetails.

2 Posesatleast2problemsandfinishessomesignificantpartsofthesolution.Communicatessomeideasbutshowsgapsintheoreticalcomprehension

1 Poses a problem but demonstrates little comprehension of how it can besolved.

Source:D.O.#73s.2012

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This lessonwasaboutthegraphicalsolutionsofsystemsof linear inequalities intwovariables.Inthislesson,youwereabletousethegraphicalmethodoffindingthesolutionsof systems of linear inequalities and given the opportunity to determine the advantagesanddisadvantagesofusingthismethod.Youwereabletofindoutwhenasystemhasnosolutionandwhenithasaninfinitenumberofsolutions.Moreimportantly,youweregiventhechancetoformulateandsolvereal-lifeproblems,makedecisionsbasedontheproblems,anddemonstrateyourunderstandingofthelessonbydoingsomepracticaltasks.

GLOSSARY OF TERMS:

1. EliminationMethod–analgebraicmethodofsolvingsystemsof linearequations. Inthismethod,thevalueofonevariableisdeterminedbyeliminatingtheothervariablethroughalgebraicmanipulation.

2. GeoGebra – an open-source dynamic mathematics software that can be used tovisualizeandunderstandconceptsinalgebra,geometry,calculus,andstatistics.

3. GraphicalMethod–amethodoffindingthesolution(s)ofasystemoflinearequationsorinequalitiesbygraphing.

4. Simultaneous linear equations or systemof linear equations – a set or collection of

linearequations,allofwhichmustbesatisfied.ItcanbewrittenasAx + By = CDx + Ey = F ,where

A,B,C,D,E,andFareallrealnumbers,andthecoefficientsofx and yarenotbothzeroforeachequation.

5. Simultaneouslinearinequalitiesorsystemoflinearinequalities–asetorcollectionoflinearinequalities,allofwhichmustbesatisfied.Thelinearinequalitiesarealsoreferredtoasconstraints.

6. Solutiontoasystemoflinearequations-thecoordinatesofallpointsofintersectionofthegraphsoftheequationsinthesystemwhosecoordinatesmustsatisfyallequationsinthesystem.

7. SubstitutionMethod–analgebraicmethodofsolvingsystemsof linearequations.Inthismethod,theexpressionequivalenttoonevariableinoneequationissubstitutedintotheotherequationtosolvefortheremainingvariable.

8. Systemofconsistentanddependentequations–isasystemoflinearequationshavinginfinitelymanysolutions.Theslopesof the linesdefinedby theequationsareequal,theiry-interceptsarealsoequal,andtheirgraphscoincide.

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9. Systemofconsistentandindependentequations–asystemoflinearequationshavingexactlyonesolution.Theslopesofthelinesdefinedbytheequationsarenotequal,theiry-interceptscouldbeequalorunequal,andtheirgraphsintersectatexactlyonepoint.

10. Systemofinconsistentequations–asystemoflinearequationshavingnosolution.Theslopesofthelinesdefinedbytheequationsareequalorbothlineshavenoslopes,theiry-interceptsarenotequal,andtheirgraphsareparallel.

REFERENCES AND WEBSITE LINKS USED IN THIS MODULE:

REFERENCES:

Bennett,JeannieM.,DavidJ.Chard,AudreyJackson,JimMilgram,JanetK.Scheer,andBertK.Waits.HoltPre-Algebra,Holt,RinehartandWinston,USA,2005.

Bernabe,JulietaG.andCecileM.DeLeon.ElementaryAlgebra,TextbookforFirstYear,JTWCorporation,QuezonCity,2002.

Brown,RichardG.,MaryP.Dolciani,RobertH.Sorgenfrey andWilliamL.Cole.Algebra,StructureandMethod,BookI,HoughtonMifflinCompany,BostonMA,1990.

Brown,RichardG.,MaryP.Dolciani,RobertH.Sorgenfrey,andRobertB.Kane.Algebra,StructureandMethodBook2.HoughtonMifflinCompany,Boston,1990.

Callanta,MelvinM.andConcepcionS.Ternida.InfinityGrade8,Worktext inMathematics.EUREKAScholasticPublishing,Inc.,MakatiCity,2012.

Chapin, Illingworth,Landau,MasingilaandMcCracken.PrenticeHallMiddleGradesMath,ToolsforSuccess,Prentice-Hall,Inc.,UpperSaddleRiver,NewJersey,1997.

Clements,DouglasH.,KennethW.Jones,LoisGordonMoseleyandLindaSchulman.MathinmyWorld,McGraw-HillDivision,Farmington,NewYork,1999.

Coxford,Arthur F. and JosephN. Payne.HBJAlgebra I, SecondEdition, Harcourt BraceJovanovich,Publishers,Orlando,Florida,1990.

Fair,JanandSadieC.Bragg.PrenticeHallAlgebraI,Prentice-Hall,Inc.,EnglewoodCliffs,NewJersey,1991.

Gantert,AnnXavier.Algebra2andTrigonometry.AMSCOSchoolPublications,Inc.,2009.

Gantert,AnnXavier.AMSCO’sIntegratedAlgebraI,AMSCOSchoolPublications,Inc.,NewYork,2007.

Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Algebra 1, Applications,Equations,andGraphs.McDougalLittell,AHoughtonMifflinCompany,Illinois,2004.

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Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Algebra 2, Applications,Equations,andGraphs.McDougalLittell,AHoughtonMifflinCompany,Illinois,2008.

Smith, Charles, Dossey, Keedy and Bettinger. Addison-Wesley Algebra, Addison-WesleyPublishingCompany,1992.

Wesner,TerryH.andHarryL.Nustad.ElementaryAlgebrawithApplications.Wm.C.BrownPublishers.IA,USA.

Wilson, Patricia S., et al.Mathematics,Applications andConnections, Course I, GlencoeDivisionofMacmillan/McGraw-HillPublishingCompany,Westerville,Ohio,1993.

WEBLINKS

A. Linear Inequalities in Two Variables

WEBSITE Links as References for Learning Activities:

1. Algebra-class.com.(2009).GraphingInequalities.RetrievedDecember11,2012,fromhttp://www.algebra-class.com/graphing-inequalities.html

2. AlgebraLabProjectManager.MainlandHighSchool.(2013).AlgebraIIRecipe:Linear Inequalities inTwoVariables.RetrievedDecember11,2012, fromhttp://algebralab.org/studyaids/studyaid.aspx?file=Algebra2_2-6.xml

3. edHelper.com. Math Worksheets. Retrieved December 11, 2012, from http://edhelper.com/LinearEquations.htm

4. HoughtonMifflinHarcourtPublishingCompany. (2008).Algebra I: Solving andGraphing Linear Inequalities. Retrieved December 11, 2012, from http://www.classzone.com/books/algebra_1/page_build.cfm?id=lesson5&ch=6

5. http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U05_L2_T1_text_final.html

6. http://www.purplemath.com/modules/ineqgrph.html

7. KGsePG. (2010). Inequalities inTwoVariables.RetrievedDecember11,2012,fromhttp://www.kgsepg.com/project-id/6565-inequalities-two-variables

8. Llarull,Marcelo.DepartmentofMathematics,WilliamPattersonUniversity.(2004).Linear Inequalities in Two Variables and Systems of Inequalities. RetrievedDecember11,2012, fromhttp://www.beva.org/maen50980/Unit04/LI-2variables.htm

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9. Monahan, Christopher. Netplaces.com. New York Times Company. GraphingLinear Inequalities inTwoVariables.RetrievedDecember11,2012, fromhttp://www.netplaces.com/algebra-guide/graphing-linear-relationships/graphing-linear-inequalities-in-two-variables.htm

10. Muhammad, Rashid Bin. Department of Computer Science, Kent University.Linear Inequalities and Linear Equations. Retrieved December 11, 2012, fromhttp://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/MathAlgor/linear.html

11. Mr.Chamberlain.FrelinghuysenMiddleSchool.(2012).SystemsofEquationsandInequalities.RetrievedDecember11,2012,fromhttp://www.mathchamber.com/algebra7/unit_06/unit_6.htm

12. Perez,Larry.(2008).BeginningAlgebra:LinearEquationsandInequalitiesinTwoVariables.RetrievedDecember11,2012,fromhttp://www.saddleback.edu/faculty/lperez/algebra2go/begalgebra/index.html#systems

13. SavannahSteele.(2008).AlgebraIResources:SystemsofLinearEquationsandInequalities. Retrieved December 11, 2012, from https://sites.google.com/site/savannaholive/mathed-308/algebra1

14. Tutorvista.com. (2010).GraphingLinearEquations inTwoVariables.RetrievedDecember 11, 2012, from http://math.tutorvista.com/algebra/equations-and-inequalities.html#

15. www.mathwarehouse.com.Linear Inequalities:How tograph theequationofalinearinequality.RetrievedDecember11,2012,fromhttp://www.mathwarehouse.com/algebra/linear_equation/linear-inequality.php

16. WyzAnt tutoring. (2012). Graphing Linear Inequalities. Retrieved December11, 2012, from http://www.wyzant.com/Help/Math/Algebra/Graphing_Linear_Inequalities.aspx

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WEBSITE Links for Videos:

1. PearsonEducation,Inc.Modelingreal-worldsituationsusinglinearinequalities.Retrieved December 11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-237s.html

2. Yahoo. (2012). Linear Inequalities in Two Variables. Retrieved December 11,2012, from http://video.search.yahoo.com/search/video?p=linear+inequalities+in+two+variables

3. Yahoo.(2012).SystemsofLinearEquationsandInequalities.RetrievedDecember11,2012,fromhttp://video.search.yahoo.com/search/video?p=systems+of+linear+equations+and+inequalities

WEBSITE Links for Images:

1. http://lazyblackcat.files.wordpress.com/2012/09/14-lex-chores-copy.png

2. http://www.google.com.ph/imgres?q=filipino+doing+household+chores&start=166&hl=fil&client=firefox-a&hs=IHa&sa=X&tbo=d&rls=org.mozilla:en-US:official&biw=1024&bih=497&tbm=isch&tbnid=e6JZNmWnlFvSaM:&imgrefurl=http://lazyblackcat.wordpress.com/2012/09/19/more-or-lex-striking-home-with-lexter-maravilla/&docid=UATH-VYeE9bTNM&imgurl=http://lazyblackcat.files.wordpress.com/2012/09/14-lex-chores-copy.png&w=1090&h=720&ei=4EC_ULqZJoG4iQfQroHACw&zoom=1&iact=hc&vpx=95&vpy=163&dur=294&hovh=143&hovw=227&tx=79&ty=96&sig=103437241024968090138&page=11&tbnh=143&tbnw=227&ndsp=17&ved=1t:429,r:78,s:100,i:238

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B. Systems of Linear Equations and Inequalities in Two Variables

WEBSITE Links as References and for L earning Activities:

1. Aprelium. (2012). Supply-AS-Sheet1.pdf .RetrievedDecember 12, 2012, fromhttp://illuminations.nctm.org/lessons/9-12/supply/Supply-AS-Sheet1.pdf

2. Aprelium. (2012). Supply-AS-Sheet2.pdf .RetrievedDecember 12, 2012, fromhttp://illuminations.nctm.org/lessons/9-12/supply/Supply-AS-sheet2.pdf

3. Coolmath.com, Inc. (2012). Solving 2 x 2 Systems of Equations. RetrievedDecember11,2012,fromhttp://www.coolmath.com/crunchers/algebra-problems-systems-equations-2x2.htm

4. Dendane,Abdelkader.UnitedArabEmiratesUniversity(2007).SolveSystemsofLinearEquations.RetrievedDecember11,2012,fromhttp://www.analyzemath.com/equations_inequalities.html

5. edHelper.com. Math Worksheets. Retrieved December 11, 2012, from http://edhelper.com/LinearEquations.htm

6. Education.com,Inc.(2012).GraphingSystemsofLinearEquationsandInequalitiesPracticeQuestions. Retrieved December 11, 2012, from http://www.education.com/study-help/article/graphing-systems-linear-equations-inequalities1/

7. Education.com,Inc.(2012).SolvingSystemsofEquationsandInequalitiesHelp.Retrieved December 11, 2012, from http://www.education.com/study- tackling-help/article/systems-equations-inequalities/

8. Flat World Knowledge. New Charter University. (2012). Elementary AlgebraChapter4.SolvingLinearSystemsbyGraphing.RetrievedDecember11,2012,fromhttps://new.edu/resources/solving-linear-systems-by-graphing

9. Flat World Knowledge. New Charter University. (2012). Elementary AlgebraChapter 4. Solving Systems of Linear Inequalities (Two Variables). RetrievedDecember 11, 2012, from https://new.edu/resources/solving-systems-of-linear-inequalities-two-variables

10. HoughtonMifflin Harcourt Publishing Company. (2008).Algebra I: Systems ofLienar Equations and Inequalities. Retrieved December 11, 2012, from http://www.classzone.com/books/algebra_1/page_build.cfm?id=none&ch=7

311

11. KGsePG. (2010). Systems of Linear Equations and Inequalities. RetrievedDecember11,2012,fromhttp://www.kgsepg.com/project-id/6653-systems-linear-equations-and-inequalities

12. +KhanAcademy.(2012).AdditionEliminationMethod2.RetrievedDecember11,2012,fromhttp://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/addition-elimination-method-2

13. LakeTahoeCommunityCollege.SolvingSystemsofEqualitiesandInequalities,andMoreWordProblems.RetrievedDecember11,2012,fromhttp://ltcconline.net/greenl/courses/152b/QuadraticsLineIneq/systems.htm

14. Ledwith,Jennifer.About.com.(2012).SystemsofLinearEquationsWorksheets.Retrieved December 11, 2012, from http://math.about.com/od/algebra1help/a/System_of_Equations_Worksheets.htm

15. Monahan, Christopher. Netplaces.com. New York Times Company. GraphingLinear Inequalities inTwoVariables.RetrievedDecember11,2012, fromhttp://www.netplaces.com/algebra-guide/graphing-linear-relationships/graphing-linear-inequalities-in-two-variables.htm

16. Monahan, Christopher. Netplaces.com. New York Times Company. Systemsof Linear Equations: Solving Graphically. Retrieved December 11, 2012, fromhttp://www.netplaces.com/algebra-guide/systems-of-linear-equations/solving-graphically.htm

17. Monahan,Christopher.Netplaces.com.NewYorkTimesCompany.SystemsofLinearEquations.RetrievedDecember11,2012,fromhttp://www.netplaces.com/algebra-guide/systems-of-linear-equations/

18. Mr.Chamberlain.FrelinghuysenMiddleSchool.(2012).SystemsofEquationsandInequalities.RetrievedDecember11,2012,fromhttp://www.mathchamber.com/algebra7/unit_06/unit_6.htm

19. Muhammad, Rashid Bin. Department of Computer Science, Kent University.Linear Inequalities and Linear Equations. Retrieved December 11, 2012, fromhttp://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/MathAlgor/linear.html

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20. Perez,Larry.(2008).BeginningAlgebra:LinearEquationsandInequalitiesinTwoVariables.RetrievedDecember11,2012,fromhttp://www.saddleback.edu/faculty/lperez/algebra2go/begalgebra/index.html#systems

21. Reger,Cheryl(2008).SystemsofEquationsandInequalities.RetrievedDecember11,2012,fromhttp://wveis.k12.wv.us/teach21/public/project/Guide.cfm?upid=3354&tsele1=2&tsele2=118

22. Stape,Elizabeth.(2012).SystemsofLinearInequalities.RetrievedDecember11,2012,fromhttp://www.purplemath.com/modules/syslneq.htm

23. Steele,Savannah.BrighamYoungUniversity(2008).SystemsofLinearEquationsand Inequalities. Retrieved December 11, 2012, from https://sites.google.com/site/savannaholive/mathed-308/algebra1

24. SOPHIA Learning, LLC. (2012) Systems of Linear Equations and InequalitiesPathway.RetrievedDecember11,2012,fromhttp://www.sophia.org/systems-of-linear-equations-and-inequalities--2-pathway

25. Tutorvista.com. (2010).GraphingLinearEquations inTwoVariables.RetrievedDecember 11, 2012, from http://math.tutorvista.com/algebra/equations-and-inequalities.html#

26. www.mathwarehouse.com.SystemsofLinearEquations.RetrievedDecember11,2012, from http://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/index.php

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WEBSITE Links for Videos:

1. Mr. Johnson’s Math Class, Hartland High School. (2012). Systems of LinearEquations and Inequalities. Retrieved December 11, 2012, from http://johnsonsmath.weebly.com/chapter-3---systems-of-linear-equations--inequalities.html

2. PearsonEducation,Inc.BreakEvenPoints.RetrievedDecember11,2012,fromhttp://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-236s.html

3. PearsonEducation,Inc.SolutionstoSystemsofEquations.RetrievedDecember11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-229s.html

4. Pearson Education, Inc. Solving Systems by Graphing. Retrieved December11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-228s.html

5. PearsonEducation,Inc.SolvingSystemsofEquationsUsingElimination.RetrievedDecember 11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-233s.html

6. PearsonEducation,Inc.SolvingSystemsUsingElimination.RetrievedDecember11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-234s.html

7. Pearson Education, Inc. Systems of Linear Inequalities. Retrieved December11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-238s.html

8. Pearson Education, Inc. Using Systems. Retrieved December 11, 2012, fromhttp://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-232s.html

9. Pearson Education, Inc. Using Systems of Inequalities. Retrieved December11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-240s.html

314

10. PearsonEducation,Inc.WritingLinearSystems.RetrievedDecember11,2012,from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-235s.html

11. Pearson Education, Inc.Writing Systems of Inequalities. Retrieved December11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-239s.html

12. YAHOO. (2012) Systems of Linear Equations and Inequalities. RetrievedDecember11,2012,fromhttp://video.search.yahoo.com/search/video?p=systems+of+linear+equations+and+inequalities

Mathematics - Gearing Towards Quality Education · 08-06-2012 · Mathematics Learner’s Module 5 Department of Education Republic of the Philippines This instructional material

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