systems of linear 5 equations and inequalities in two ...€¦ · lesson 3 – graphical solutions...

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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

5

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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 • Graphsystemoflinearinequalitiesintwovariables.• 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

3y – 7 = 11

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.Shewasaskedfurthertopresenttotheclubhowshecameupwiththeresultusinggraph.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: Show the graph of each of the following linear equations in aCartesiancoordinateplane.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,seektheassistanceofyourteacherorpeersorrefertothemodulesyouhavegoneoverearlier.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: Show the graph of each pair of linear equations below using the sameCartesianplanethenanswerthequestionsthatfollow.

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

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3. x + 3y = 6 and 2x + 6y = 12

QU

ESTIONS?

a. Howdidyougrapheachpairoflinearequations?b. Howwouldyoudescribethegraphsof3x + y = 5 and 2x + y=9? Howabout3x – y = 4 and y = 3x+2?x + 3y = 6 and 2x + 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 = 6 and 2x + 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 equationsifwewantthemtobetrueforthesamepairofnumbers.Thesolutionofsuchequationsisanorderedpairofnumbersthatsatisfiesbothequations.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. onlyonesolutioniftheirgraphsintersect. b. nosolutioniftheirgraphsdonotintersect. c. infinitelymanysolutionsiftheirgraphscoincide.

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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

definedbytheequationsareequalorhavenoslopes,theiry-interceptsarenotequal;andtheirgraphsareparallel.

Example: Thesystemofequations 2x + y = -6 2x + y =10 is inconsistent.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, consistent and independent, or inconsistent. Answer thequestionsthatfollow.

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. Howwereyouabletoidentifysystemofequationsthatareconsistent-dependent,consistent-independentandinconsistent?

b. Whendoyousaythatasystemoflinearequationsisconsistentanddependent?consistentandindependent?inconsistent?

c. Giveexamplesofsystemsoflinearequationsthatareconsistentanddependent,consistentandindependent,andinconsistent.

Were you able to determine which system of linear equations in two variables is 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: Describethesolutionsetofthesystemoflinearequationsasshownbythefollowinggraphs.Answerthequestionsthatfollow.

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 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.Answerthequestionsthatfollow.

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

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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.

What to UnderstandWhat to Understand

Yourgoal inthissectionistotakeacloser lookatsomeaspectsofthetopic.Youaregoingtothinkdeeperandtestfurtheryourunderstandingofsystemsoflinearequations in twovariablesandtheirgraphs.Afterdoingthefollowingactivities,youshouldbeabletoanswerthefollowingquestion:Howisthesystemoflinearequationsintwovariablesusedinsolvingreal-lifeproblemsandinmakingdecisions?

HOW WELL I UNDERSTOOD…Activity 6

Directions: Answerthefollowing.

1. Howdoyoudescribeasystemoflinearequationsintwovariables?2. Give at least two examples of systems of linear equations in two

variables. 3. Whenisasystemoflinearequationsintwovariablesused? 4. Howdoyougraphsystemsoflinearequationsintwovariables?

5. Howdoyoudescribethegraphsofsystemsoflinearequationsintwovariables?

6. Howdoyoudescribesystemsoflinearequationsthatareconsistentanddependent?consistentandindependent?inconsistent?

7. Studythesituationonthenextpage:

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a. Whatsystemoflinearequations

representsthegivensituation?b. Suppose the system of linear

equations is graphed. Howwouldthegraphslooklike?

c. Is the system consistent anddependent, consistent andindependent, or inconsistent?Why?

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

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

Josewantedtoconstructarectangular

gardensuchthatitsperimeteris28manditslengthis6timesitswidth.

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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 1Systematicallylistedinthetablethedata,properlyformulatedlinearequationsintwovariablesthatformasystemofequations,andaccuratelydrawnthegraphofeachsystemoflinearequations.

Systematicallylistedinthetablethedata,properlyformulatedlinearequationsintwovariablesthatformasystemofequationsbutunabletodrawthegraphaccurately.

Systematicallylistedinthetablethedataandformulatedlinearequationsintwovariablesbutunabletoformsystemsofequations.

Systematicallylistedinthetablethedata.

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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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,seektheassistanceofyourteacherorpeersorrefertothemodulesyouhavegoneoverearlier.

HOW MUCH IS THE FARE?Activity 1

Directions: Usethesituationbelowtoanswerthequestionsthatfollow.

Suppose for a given distance, a tricycle driver charges Php 10.00everypassengerwhileajeepneydriverchargesPhp12.00.

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. Supposein3roundtripsthetricycleandjeepneydrivershadhadatotalof68passengers.

a. Howwouldyoufindthenumberofpassengerseachhad?b. Whatmathematicalstatementwillyouusetofindthenumberof

passengerseachhad?

Whatisthetotalamountoffarecollectedfromthepassengersbythetwodrivers?Explainhowyouarrivedatyouranswer.

c. Howwouldyoudrawthegraphofthemathematicalstatementobtainedin3b?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: Usethesituationbelowtoanswerthequestionsthatfollow. 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.

The solution of a system of linear equations can be determined algebraically orgraphically.Tofindthesolutiongraphically,graphbothequationsonaCartesiancoordinateplanethenfindthepointofintersectionofthegraphs,ifitexists.YoumayalsousegraphingcalculatororcomputersoftwaresuchasGeoGebraindeterminingthegraphicalsolutionsofsystemsoflinearequations.GeoGebraisadynamicmathematicssoftwarewhichhelpsyouvisualizeandunderstandconceptsinalgebra,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): The graphs of 3x + y = 4

and 3x + y=10areparallel.

Hence,thesystem3x + y = 43x – y = -5

hasnosolution.

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. Substitutetothesecondequationthevalueofthevariablefoundinthefirststep.

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 = 6

2 y = 3

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Checkthevaluesofthevariablesobtainedagainstthelinearequationsinthesystem.

1. 2x + y=5; x = 1 and y = 3 2(1)+3=5 2 + 3 = 5 5 = 5 Hence,x = 1 and y=3aretrueto2x + y=5.

2. -x + 2y=5; x = 1 and y = 3 -1+2(3)=5 -1 + 6 = 5 5 = 5 Hence,x = 1 and y=3aretrueto-x+2y=5

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 = 16 17x = 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)=7 9 – 2 = 7 7 = 7 Hence,x = 3 and y=-2aretrueto3x + y=7.

2. 2x – 5y=16; x = 3 and y = -2 2(3)–5(-2)=16 6+10=16 16 = 16 Hence,x=3andy=-2aretrueto2x – 5y=16.

Therefore,thesolutiontothesystem3x + 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.

Findthedailywageofasupervisorbysubstituting450toxinanyoftheoriginalequations.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.

Youhaveseenhowsystemoflinearequationsisusedtosolvereal-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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Were you able to find the solution set of each system of linear equations? Do you think this is the most convenient way of solving a system of equations? In the next activity, you will be determining the number(s) that must be multiplied to the terms of one or both equations in a system of equations. This will lead you in 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 resulting equation by substituting the given value of onevariabletoeachofthefollowingequations.Thensolvefortheothervariableusingtheresultingequation.Answerthequestionsthatfollow.

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. Whichsystemofequationsisdifficulttosolve?Why?d. Whichsystemofequationshasnosolution?Why?e. Whichsystemofequationshasinfinitenumberofsolutions?Explain

youranswer.

ELIMINATE ME!Activity 8

Directions: Determinethenumber(s)thatmustbemultipliedtooneorbothequationsineachsystemtoeliminateoneofthevariables.Justifyyouranswer.

1. 5x – 2y = 122x + y = 7

2. x + 3y = 54x + 2y = 7

To eliminate x To eliminate y

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3. x – 4y = 125x + y = -5

4. -3x + 2y = 75x – 4y = -2

5. -2x – 5y = 103x – 2y = 6

6. 9x – 5y = 83x + 7y = 12

To eliminate x To eliminate y

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 elimination method.

ELIMINATE THEN SOLVE!Activity 9

Directions: Solveeachsystemof linearequationsbyeliminationmethodthencheck.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. 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.

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In this section, the discussion was about solving systems of linear equations in two variables using the graphical and algebraic methods.


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.

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. Howwouldyoucheckifthesolutionsetyoufoundfromthegraphsoflinearequationsinasystemarethesolutions?

5. Whatdoyouthinkaretheadvantagesandthedisadvantagesofthegraphicalmethodofsolvingsystemsoflinearequations?Explainyouranswer.

Were you able to answer all the questions in the activity? Do you have better understanding now 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 WORK…Activity 11

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

1. Howdoyoudescribeeachequationinthesystem?2. Howwillyousolvethegivensystemofequations?3. Do you think the substitutionmethod ismore convenient to use in

findingthesolutionsetofthesystem?Explainyouranswer.4. Whatisthesolutionsetofthegivensystemofequations?Explainhow

youarrivedatyouranswer.5. Whenisthesubstitutionmethodconvenienttouseinsolvingsystems

oflinearequations?6. Give two examples of systemsof linear equations in two variables

thensolveusingthesubstitutionmethod.

How did you find the activity? Were you able to have 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 opportunities to deepen your understanding of solving systems of linear equations in two variables using the elimination method. You were able to find out also systems of linear equations that can be solved conveniently by using substitution or elimination method. In the next activity, you will extend your understanding of systems of linear equations in two variables as 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. Howdoyoudescribeeachequationinthesystem?2. Howwillyousolvethegivensystemofequations?3. Whichalgebraicmethodofsolvingsystemoflinearequationsdoyou

thinkismoreconvenienttouseinfindingitssolutionset?Why?4. Whatisthesolutionsetofthegivensystemofequations?Explainhow

youarrivedatyouranswer.5. Whenistheeliminationmethodconvenienttouseinsolvingsystems

oflinearequations?6. Give two examples of systems of linear equations in two variables

thensolveusingtheeliminationmethod.

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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. Luisasells twobrandsofTabletPC.Shereceivesacommissionof12%ononebrandand8%ontheotherbrand.Ifsheisabletosell2TabletPC's,oneforeachbrand,thetotalcostisPhp42,000andtheamountthatshewillreceiveascommissionisPhp4,400.

a. WhatisthecostofeachbrandofTabletPCs? b. Howmuchcommissiondoesshereceiveononebrand? Howaboutontheotherbrand?

c. SupposeyouareLuisaandyouwishtoearnmore.WhichTabletPCwillyouaskthecustomersorclientstobuy?Why?

3. Whichofthefollowingmobilenetworkshasabetteroffer?Justifyyouranswer.

WorldCelcom: Php500monthlycharge FreecallsandtextstoWorldCelcomsubscribers Php6.50perminuteofcalltoothernetworks

Smartlink: Php650monthlycharge FreecallsandtextstoSmartlinksubscribers Php5perminuteofcalltoothernetworks

4. Mr.Salongahastwoinvestments.HistotalinvestmentisPhp400,000.Hereceives3%interestonone investmentand7%interestontheother.Thetotal interestthatMr.SalongareceivesinayearisPhp16,000.

a. HowmuchmoneydoesMr.Salongahaveineachinvestment?b. Suppose youwereMr.Salonga. Inwhich investmentwill you

placemoremoney?Why?

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5. Theschoolcanteensellschickenandeggsandwiches.Itgeneratesan incomeofPhp2 foreverychickensandwichsoldandPhp1.25for every egg sandwich. Yesterday, they were able to sell all 420sandwiches prepared and generated an income of Php 615. Theteacherinchargeofthecanteenrealizedthatthecanteencouldhaveearnedmoreifadditionalsandwichesareprepared.

a. Howmanychickensandwicheswasthecanteenabletosellonthatday?Howabouteggsandwiches?

b. Ifyouweretheteacherinchargeofthecanteen,whichkindofsandwichwouldyoupreparemore?Why?

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.


Yourgoalinthissectionistoapplyyourlearningtoreal-lifesituations.Youwillbegivenapracticaltaskinwhichyouwilldemonstrateyourunderstandingofsolvingsys-temsoflinearequationsintwovariables.

PLAY THE ROLE OF …Activity 14

Citesituationsinreallifewheresystemsoflinearequationsintwovariablesareapplied.Formagroupof5membersandroleplayeachsituation.Withyourgroupmates,formulateproblemsoutofthesesituationsthensolveinasmanywaysasyoucan.

SELECT THE BEST POSTPAID PLANActivity 15

1. Make a list of all postpaid plans being offered by different mobile networkcompanies.

2. Usethepostpaidplanstoformulateproblemsinvolvingsystemsoflinearequationsintwovariables.Solvetheproblemsformulated.Usetherubricprovidedtorateyourwork.

3. Determinethebestpostpaidplanthateachcompanyoffers.Explainyouranswer.4. Determinethemobilenetworkcompanythatyouwillrecommendtoyourparents,

olderbrothersorsisters,orrelativesifevertheyapplyforapostpaidplan.Justifyyourchoice.

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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?

RubriconProblemsFormulatedandSolvedScore Descriptors

6 Poses amore complex problemwith 2 ormore correct possible solutionsand 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.

267

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,shesellssouvenir itemssuchasbraceletsandnecklaceswhicharemadeof localshells.EachbraceletcostsPhp85whileeachpieceofnecklaceisPhp110.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’stotalsaleifshesells5piecesofbraceletsand5piecesofnecklaces?

How about if she sells 10 pieces of bracelets and 20 pieces ofnecklaces?

3. What mathematical statement would represent the total sale ofbraceletsandnecklaces?Describethemathematicalstatementthengraph.

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

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 show the graphs of systems of linear equations and inequalities in two variables. You need this skill to learn about the graphical solutions of systems of linear inequalities in two variables.

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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? How about 5x – y = 12 and 5x – y > 12? 2x + 3y = 15 and 2x + 3y≤15?c. Howdoyoudifferentiatethegraphsoflinearequationsandinequalities

intwovariables?d. Howmanysolutionsdoesa linearequation in twovariableshave?

Howaboutlinearinequalitiesintwovariables?e. Suppose you draw the graphs of3x + y < 10and 5x – y > 12 on

anotherCartesian coordinate plane.Howwould youdescribe theirgraphs?Whatorderedpairssatisfybothinequalities?

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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 determine the ordered pairs that satisfy two linear inequalities? Finding solutions of linear inequalities leads you in understanding the graphical solutions of systems of linear inequalities in two variables. But how are systems of linear inequalities 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. To prepare yourself in performing these activities, read and understand first some important notes on the Graphical Solutions of Systems of Linear Inequalities in Two Variables and the examples presented.

Tosolveasystemofinequalitiesintwovariablesbygraphing,drawthegraphofeachinequalityon thesame rectangularcoordinateplane.Each time,shade thesolutionsetofeachinequality.Thesolutionsetofthesystemistheregionwheretheshadingsoverlap.

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

thesameCartesiancoordinateplane.Theregionwheretheshadingsoverlapisthegraphofthesolutiontothesystem.

Example: Thereareatmost56peoplecomposedofchildrenandadultswhoareinabus.EachchildandadultpaidPhp80andPhp100,respectively.Ifthetotalamount collectedwasnotmore thanPhp4,800,howmanychildrenandadultsareinthebus?

Solution: Letx=numberofchildreninthebus y=numberofadultsinthebus

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

Like systems of linearequationsintwovariables,systemsoflinearinequalitiesarealsoappliedin many real-life situations. Theyareusedtorepresentsituationsandsolve problems related to uniformmotion, mixture, investment, work,andmanyothers.

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Usethetwoinequalitiestofindthenumberofchildrenandadultswhoareinthebus.Writetheseasasystemoflinearinequalitiesthensolvegraphically.

The regionwhere theshadingsoverlap is thegraphof thesolution

tothesystem.Consideranypointinthisshadedregionthensubstituteitscoordinatesinthesystemtocheck.

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)≤4,800or1,600+3,000≤4,800or4,600≤4,800.

Thesecondinequalityisalsosatisfied.Thismeansthatonepossible

numberofchildreninthebusis20andthenumberofchildrenis30.

However,notallpointsintheregionwheretheshadingsoverlaparesolutionstothegivensituation.Onlythosevaluesofxgreaterthanorequaltozero(x≥0)andthosevaluesofygreaterthanorequaltozero(x≥0)canonlybeconsidered.Canyouthinkofthereason?Definitely,thenumberofchildrenandadultscanneverbenegative.

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Youhaveseenhowsystemoflinearinequalitiesintwovariablesisusedtosolvereal-lifeproblem.Inwhatotherreal-lifesituationsaresystemsof linear inequalities in twovariables illustratedorapplied?Howisthesystemoflinearinequalitiesintwovariablesusedinsolvingreal-lifeproblemsandinmakingdecisions?


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


Your goal in this section is to learn and understand solving systems of linearinequalitiesintwovariablesgraphically.Usethemathematicalideasandtheexamplespresentedinansweringtheactivitiesprovided.

DO I SATISFY YOU?Activity 3

Directions: Determineifeachorderedpairisasolutiontothesystemoflinearinequality2x + 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. Howdidyouknow if thegivenorderedpair isnotasolutionof thesystem?

c. Howmanysolutionsdoyouthinkthesystemofinequalitieshas?

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Were you able to find out if each ordered pair is a solution 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.

AM I IN THAT REGION?Activity 4

Directions: Solvethefollowingsystemsofinequalitiesgraphicallythengivethreeorderedpairs satisfying the inequalities. Show that the ordered pairs satisfy theinequalities.Answerthequestionsthatfollow.Thefirstoneisdoneforyou.

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).

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5.x + y < 12y < -3x + 5 8.

2x – y≥102y≥5x + 1

6.y > 2x + 72x – y < 12 9.

2x – y < 113x + 5y≥8

7.x + 3y > 9x – 3y≤9 10.

6x + 2y≥93x + y≤-6

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QU

ESTIONS?

a. How did you determine the graphical solutions of each system oflinearinequalitiesintwovariables?

b. Howdidyouknowthattheorderedpairsyoulistedaresolutionsofthesystemofinequalities?

c. Whichsystemoflinearinequalitieshasnosolution?Why?d. Whendoyousaythatasystemoflinearinequalitieshassolutions?

hasnosolutions?e. Give 2 examples of system of linear inequalities in two variables

havingnosolutions.Justifyyouranswer.

In this section, the discussion was about solving systems of linear inequalities in two variables graphically.


Now that you know the important ideas about the graphical solutions of systems of linear inequalities in two variables, let’s go deeper by moving on to 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?”

REGION IN A PLANE Activity 5

Directions: Answerthefollowingquestions.1. Showthegraphofthesolutionofthesystem

2x + 5y < 153x – y≥8 .Usethe

Cartesiancoordinateplaneonthenextpage.

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

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

15 and 3x – y≥8meet.Whatstatementscanyoumakeabout thecoordinatesofthesepoints?

5. How would you describe the graphical solutions of the system2x + 5y < 153x – y≥8

?

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

Howisitsimilarordifferentfromthegraphicalsolutionofsystemoflinearequations?

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

LOOKING CAREFULLY AT THE REGION…Activity 6

Directions: Answerthefollowingquestions.

1. Howdoyoudeterminethesolutionsetofasystemoflinearinequalitiesintwovariablesfromitsgraph?

2. Doyou think it iseasy todetermine thesolutionsetofasystemoflinearinequalitiesbygraphing?Explainyouranswer.

3. Inwhatinstancewillyoufinditdifficulttodeterminethesolutionsetofasystemoflinearinequalitiesfromitsgraph?

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The activity provided you opportunities to deepen your understanding of solving systems of linear inequalities in two variables graphically. In the next activity, you will extend your understanding of systems of linear inequalities in two variables as to how they are used in solving real-life problems and in making decisions.

4. Howwouldyouknow if thesolutionsyou found from thegraphsoflinearinequalitiesinasystemaretrue?

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. TicketsinaplaycostPhp250foradultsandPhp200forchildren.ThesponsoroftheshowcollectedatotalamountofnotmorethanPhp44,000frommorethan150adultsandchildrenwhowatchedtheplay.

a. Whatmathematicalstatementsrepresentthegivensituation?b. Drawanddescribethegraphsofthemathematicalstatements.c. Howwillyoufindthenumberofchildrenandadultswhowatched

theplay?d. Give4possiblenumbersofadultsandchildrenwhowatchedthe

play.Justifyyouranswer.e. Thesponsoroftheshowrealizedthatifthepricesofthetickets

werereduced,morepeoplewouldhavewatchedtheplay.Ifyouwerethesponsoroftheplay,wouldyoureducethepricesofthetickets?Why?

2. Mr.Agoncillohassavingsaccountintwobanks.ThecombinedamountofthesesavingsisatleastPhp150,000.Onebankgivesaninterestof4%whiletheotherbankgives6%.Inayear,Mr.AgoncilloreceivesatmostPhp12,000.

a. Whatmathematicalstatementsrepresentthegivensituation?b. Drawanddescribethegraphsofthemathematicalstatements.c. Howwill you determine the amount of savings in each bank

account?

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d. Give 4 possible amounts of savings in both accounts. Justifyyouranswer.

e. IfyouwereMr.Agoncillo,inwhatbankaccountwouldyouplacegreateramountofmoney?Why?

3. Mrs.Burgoswantstobuyatleast30kilosofporkandbeefforherrestaurant

businessbuthastospendnomorethanPhp12,000.AkiloofporkcostsPhp180andakiloofbeefcostsPhp220.

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

Burgosneedstobuy?d. Give 4 possible amounts of pork and beef that Mrs. Burgos

needstobuy.Justifyyouranswer.e. Mrs.Burgosobserved thateveryweek, thenumberofpeople

comingtoherrestaurantisincreasing.Shedecidedtobuymoreporkandbeef tomeet thedemandsof her customers. If youwereMrs.Burgos,wouldyoudothesame?Why?

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 in real life what you have learned by doing the tasks in the next section.


Yourgoalinthissectionistoapplyyourlearningtoreal-lifesituations.Youwillbegivenapracticaltaskwhichwilldemonstrateyourunderstandingofthegraphicalsolu-tionsofsystemsoflinearinequalitiesintwovariables.

PLAY THE ROLE OF …Activity 8

Cite situations in real life where systems of linear inequalities in two variables areapplied.Formagroupof5membersand roleplayeachsituation.Withyourgroupmates,formulateproblemsoutofthesesituationsthensolveinasmanywaysasyoucan.

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How did you find the performance task? How did the task help you see the real-world applications of systems of inequalities in two variables? What important things have you learned in the activity? What values are being practiced?

JOIN THE CAMP!Activity 9

Directions: Performthefollowingactivity.Refertothesituationbelow.

Youareoneof themembersof theBoysScoutsof thePhilippinesinyourschoolwhowillbejoiningtheNationalJamboreenextmonth.Yourscoutmasterassignedyoutogetherwithyourtroopmemberstotakechargeofallthecampingmaterialsneededsuchastents,ropes,bamboos,cookingutensils, fire woods, and other necessary materials. He also asked youto prepare the foodmenu for the duration of the jamboree including theingredients.

1. Makealistofallcampingmaterialsneededincludingthequantityofeach.2. Usethecampingmaterialsandtheirquantitiestoformulateproblems

involving systems of linear inequalities in two variables. Solve theproblemsformulated.Usetherubricprovidedtorateyourwork.

3. Determineifthecampingmaterialsneededareenoughforthenumberofboysscoutswhowilljointhejamboree.Explainyouranswer.

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 Posesamorecomplexproblemandfinishesallsignificantpartsofthesolutionandcommunicatesideasunmistakably,showsin-depthcomprehensionofthepertinentconceptsand/orprocesses.

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

3 Posesacomplexproblemandfinishesmostsignificantpartsofthesolutionand communicates ideas unmistakably, shows comprehension of majorconceptsalthoughneglectsormisinterpretslesssignificantideasordetails.

2 Posesaproblemandfinishessomesignificantpartsofthesolutionandcom-municatesideasunmistakablybutshowsgapsontheoreticalcomprehension.

1 Posesaproblembutdemonstratesminorcomprehension,notbeingabletodevelopanapproach.

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 advantagesand disadvantages of using suchmethod.Using thismethod of solving systemsof linearinequalities, you were able to find out which system has no solution and infinite numberof solutions.More importantly, youweregiven the chance to formulateand solve real-lifeproblems,makedecisionsbasedontheproblems,anddemonstrateyourunderstandingofthelessonbydoingsomepracticaltasks.

Glossary of Terms:

1. EliminationMethod–Thisisanalgebraicmethodofsolvingsystemsoflinearequations.Inthismethod,thevalueofonevariableisdeterminedbyeliminatingtheothervariable.Toeliminatethevariable,somemathematicaloperationsarefollowed.

2. GeoGebra–Thisisadynamicmathematicssoftwarethatcanbeusedtovisualizeandunderstandconceptsinalgebra,geometry,calculus,andstatistics.

3. GraphicalMethod–This isamethodof finding thesolution(s)ofasystemof linearequationsbygraphing.

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

5. Simultaneouslinearinequalitiesorsystemoflinearinequalities–asetorcollectionofinequalitiesthatonesolvesalltogetheratonce.

6. Solutiontoasystemof linearequations-Thiscorrespondstothecoordinatesof thepointsofintersectionofthegraphsoftheequations.

7. SubstitutionMethod–Thisisanalgebraicmethodofsolvingsystemsoflinearequations.Inthismethod,theexpressionequivalenttoonevariableinoneequationissubstitutedtotheotherequationtosolvefortheothervariable.

8. Systemofconsistentanddependentequations–Thisisasystemoflinearequationshavinginfinitelymanysolutions.Theslopesofthelinesdefinedbytheequationsareequal,theiry-interceptsarealsoequal,andtheirgraphscoincide.

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

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10. System of inconsistent equations – This is a system of linear equations having nosolution.Theslopesofthelinesdefinedbytheequationsareequalorhavenoslopes,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.AlementaryAgebra,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.

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

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

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

Wilson,PatriciaS., et. al.Mathematics,ApplicationsandConnections,Course I,GlencoeDivisionofMacmillan/McGraw-HillPublishingCompany,Westerville,Ohio,1993.

WEBLINKS

A. Linear Inequalities in Two Variables

WEBSITE Links as References and for Learning Activities:

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

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

3. edHelper.com.MathWorksheets.RetrievedDecember11,2012,fromhttp://edhelper.com/LinearEquations.htm

4. HoughtonMifflinHarcourtPublishingCompany.(2008).AlgebraI:SolvingandGraphingLinear 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.html7. KGsePG. (2010). Inequalities inTwoVariables.RetrievedDecember 11, 2012, from

http://www.kgsepg.com/project-id/6565-inequalities-two-variables8. Llarull, Marcelo. Department of Mathematics, William Patterson University. (2004).

LinearInequalitiesinTwoVariablesandSystemsofInequalities.RetrievedDecember11,2012,fromhttp://www.beva.org/maen50980/Unit04/LI-2variables.htm

9. Monahan, Christopher. Netplaces.com. NewYork Times Company. Graphing LinearInequalitiesinTwoVariables.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. LinearInequalities and Linear Equations. Retrieved December 11, 2012, from http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/MathAlgor/linear.html

11. Mr. Chamberlain.Frelinghuysen Middle School. (2012). Systems of Equations andInequalities. Retrieved December 11, 2012, from http://www.mathchamber.com/algebra7/unit_06/unit_6.htm

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12. Perez, Larry. (2008). Beginning Algebra: Linear Equations and Inequalities in TwoVariables. Retrieved December 11, 2012, from http://www.saddleback.edu/faculty/lperez/algebra2go/begalgebra/index.html#systems

13. Savannah Steele. (2008). Algebra I Resources: Systems of Linear Equations andInequalities. Retrieved December 11, 2012, from https://sites.google.com/site/savannaholive/mathed-308/algebra1

14. Tutorvista.com. (2010). Graphing Linear Equations in Two Variables. RetrievedDecember11,2012,fromhttp://math.tutorvista.com/algebra/equations-and-inequalities.html#

15. www.mathwarehouse.com.Linear Inequalities:Howtograph theequationofa linearinequality.RetrievedDecember11,2012,fromhttp://www.mathwarehouse.com/algebra/linear_equation/linear-inequality.php

16. WyzAnttutoring.(2012).GraphingLinearInequalities.RetrievedDecember11,2012,fromhttp://www.wyzant.com/Help/Math/Algebra/Graphing_Linear_Inequalities.aspx

WEBSITE Links for Videos:

1. Pearson Education, Inc. Modeling real-world situations using linear inequalities.Retrieved December 11, 2012, from http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-237s.html

2. Yahoo.(2012).LinearInequalitiesinTwoVariables.RetrievedDecember11,2012,fromhttp://video.search.yahoo.com/search/video?p=linear+inequalities+in+two+variables

3. Yahoo.(2012).SystemsofLinearEquationsandInequalities.RetrievedDecember11,2012, from http://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.png2. 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 .RetrievedDecember12,2012, fromhttp://illuminations.nctm.org/lessons/9-12/supply/Supply-AS-Sheet1.pdf

2. Aprelium. (2012).Supply-AS-Sheet2.pdf .RetrievedDecember12,2012, fromhttp://illuminations.nctm.org/lessons/9-12/supply/Supply-AS-sheet2.pdf

3. Coolmath.com,Inc.(2012).Solving2x2SystemsofEquations.RetrievedDecember11,2012,fromhttp://www.coolmath.com/crunchers/algebra-problems-systems-equations-2x2.htm

4. Dendane, Abdelkader. United Arab Emirates University (2007). Solve Systems ofLinearEquations.RetrievedDecember11,2012, from http://www.analyzemath.com/equations_inequalities.html

5. edHelper.com.MathWorksheets.RetrievedDecember11,2012,fromhttp://edhelper.com/LinearEquations.htm

6. Education.com, Inc. (2012).GraphingSystems of Linear Equations and InequalitiesPractice Questions. Retrieved December 11, 2012, from http://www.education.com/study-help/article/graphing-systems-linear-equations-inequalities1/

7. Education.com, Inc. (2012). Solving Systems of Equations and Inequalities Help.Retrieved December 11, 2012, from http://www.education.com/study- tackling- help/article/systems-equations-inequalities/

8. FlatWorldKnowledge.NewCharterUniversity.(2012).ElementaryAlgebraChapter4.SolvingLinearSystemsbyGraphing.RetrievedDecember11,2012,fromhttps://new.edu/resources/solving-linear-systems-by-graphing

9. FlatWorldKnowledge.NewCharterUniversity.(2012).ElementaryAlgebraChapter4.SolvingSystemsofLinearInequalities(TwoVariables).RetrievedDecember11,2012,fromhttps://new.edu/resources/solving-systems-of-linear-inequalities-two-variables

10. HoughtonMifflinHarcourtPublishingCompany. (2008).Algebra I:SystemsofLienarEquationsandInequalities.RetrievedDecember11,2012,fromhttp://www.classzone.com/books/algebra_1/page_build.cfm?id=none&ch=7

11. KGsePG.(2010).SystemsofLinearEquationsandInequalities.RetrievedDecember11, 2012, from http://www.kgsepg.com/project-id/6653-systems-linear-equations-and-inequalities

12. +KhanAcademy.(2012).AdditionEliminationMethod2.RetrievedDecember11,2012,from http://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

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14. Ledwith, Jennifer. About.com. (2012). Systems of Linear Equations Worksheets.RetrievedDecember11,2012,fromhttp://math.about.com/od/algebra1help/a/System_of_Equations_Worksheets.htm

15. Monahan, Christopher. Netplaces.com. NewYork Times Company. Graphing LinearInequalitiesinTwoVariables.RetrievedDecember11,2012,fromhttp://www.netplaces.com/algebra-guide/graphing-linear-relationships/graphing-linear-inequalities-in-two-variables.htm

16. Monahan,Christopher.Netplaces.com.NewYorkTimesCompany.SystemsofLinearEquations: Solving Graphically. Retrieved December 11, 2012, from http://www.netplaces.com/algebra-guide/systems-of-linear-equations/solving-graphically.htm

17. Monahan,Christopher.Netplaces.com.NewYorkTimesCompany.SystemsofLinearEquations. Retrieved December 11, 2012, from http://www.netplaces.com/algebra-guide/systems-of-linear-equations/

18. Mr. Chamberlain.Frelinghuysen Middle School. (2012). Systems of Equations andInequalities. Retrieved December 11, 2012, from http://www.mathchamber.com/algebra7/unit_06/unit_6.htm

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

20. Perez, Larry. (2008). Beginning Algebra: Linear Equations and Inequalities in TwoVariables. Retrieved December 11, 2012, from http://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. Brigham Young University (2008). Systems of Linear Equationsand Inequalities. Retrieved December 11, 2012, from https://sites.google.com/site/savannaholive/mathed-308/algebra1

24. SOPHIALearning,LLC.(2012)SystemsofLinearEquationsandInequalitiesPathway.RetrievedDecember11,2012,fromhttp://www.sophia.org/systems-of-linear-equations-and-inequalities--2-pathway

25. Tutorvista.com. (2010). Graphing Linear Equations in Two Variables. RetrievedDecember11,2012,fromhttp://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’sMathClass,HartlandHighSchool.(2012).SystemsofLinearEquationsandInequalities.RetrievedDecember11,2012,fromhttp://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. PearsonEducation,Inc.SolvingSystemsbyGraphing.RetrievedDecember11,2012,fromhttp://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-228s.html

5. PearsonEducation, Inc. SolvingSystems of EquationsUsingElimination.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. PearsonEducation,Inc.SystemsofLinearInequalities.RetrievedDecember11,2012,fromhttp://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-238s.html

8. PearsonEducation,Inc.UsingSystems.RetrievedDecember11,2012,fromhttp://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-232s.html

9. PearsonEducation,Inc.UsingSystemsofInequalities.RetrievedDecember11,2012,fromhttp://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-240s.html

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

11. PearsonEducation,Inc.WritingSystemsofInequalities.RetrievedDecember11,2012,fromhttp://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-239s.html

12. YAHOO.(2012)SystemsofLinearEquationsandInequalities.RetrievedDecember11,2012, from http://video.search.yahoo.com/search/video?p=systems+of+linear+equations+and+inequalities

systems of linear 5 equations and inequalities in two ...€¦ · lesson 3 – graphical solutions...

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