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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
Summary/Synthesis/Generalization ............................................................... 289
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
Summary/Synthesis/Generalization ............................................................... 305
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.
260
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.
262
What I have learned so far...________________________
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_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________.
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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REFLECTIONREFLECTION
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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
What to KnowWhat to Know
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?
LearnmoreaboutSystemsofLinearEquationsinTwoVariablesandtheirGraphsthroughtheWEB.Youmayopenthefollowinglinks.
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
5. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-229s.html
6. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-232s.html
7. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-233s.html
8. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-234s.html
9. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-235s.html
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.
What to ProcessWhat to Process
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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REFLECTIONREFLECTION
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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.
What to TransferWhat to Transfer
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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In this lesson, I have understood that ______________
________________________________________________
________________________________________________
_________________________________________________
________________________________________________
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________________________________________________
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___________.
REFLECTIONREFLECTION
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SUMMARY/SYNTHESIS/GENERALIZATION:
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
What to KnowWhat to Know
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.
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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? 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.
LearnmoreaboutSystemsofLinearEquationsinTwoVariablesandtheirGraphsthroughtheWEB.Youmayopenthefollowinglinks.
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
4. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-238s.html
5. h t t p : / /www.phschoo l . com/atschool/academy123/english/academy123_content/wl-book-demo/ph-240s.html
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.
What to ProcessWhat to Process
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.
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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
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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.
What to TransferWhat to Transfer
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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In this lesson, I have understood that ______________
________________________________________________
________________________________________________
_________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
_________________________________________________
________________________________________________
_________________________________________________
________________________________________________
___________.
REFLECTIONREFLECTION
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SUMMARY/SYNTHESIS/GENERALIZATION:
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