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PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (CORE) Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India CENTRAL BOARD OF SECONDARY EDUCATION CENTRAL BOARD OF SECONDARY EDUCATION CLASS X Unit 3
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PAIR OF LINEAR EQUATIONS - NIMS Dubai of Linear Equations in two variables (Core) Graphical representation of linear Equations Plotting the lines representing two linear equations

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  • PAIR OF

    LINEAR EQUATIONS IN TWO VARIABLES

    (CORE)

    Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 IndiaShiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

    CENTRAL BOARD OF SECONDARY EDUCATIONCENTRAL BOARD OF SECONDARY EDUCATION

    CLASS

    XUnit 3

  • The CBSE-International is grateful for permission to reproduce and/or

    translate copyright material used in this publication. The

    acknowledgements have been included wherever appropriate and

    sources from where the material has been taken duly mentioned. In

    case anything has been missed out, the Board will be pleased to rectify

    the error at the earliest possible opportunity.

    All Rights of these documents are reserved. No part of this publication

    may be reproduced, printed or transmitted in any form without the

    prior permission of the CBSE-i. This material is meant for the use of

    schools who are a part of the CBSE-International only.

  • The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos.

    The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view.

    The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary.

    The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements.

    The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners.

    The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some non-evaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve.

    The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens.

    The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board.

    I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the development and implementation of this material.

    The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions are welcome.

    Vineet JoshiChairman

    PrefacePreface

  • AcknowledgementsAcknowledgements

    English :

    Geography:

    Ms. Sarita Manuja

    Ms. Renu Anand

    Ms. Gayatri Khanna

    Ms. P. Rajeshwary

    Ms. Neha Sharma

    Ms. Sarabjit Kaur

    Ms. Ruchika Sachdev

    Ms. Deepa Kapoor

    Ms. Bharti Dave Ms. Bhagirathi

    Ms. Archana Sagar

    Ms. Manjari Rattan

    Mathematics :

    Political Science:

    Dr. K.P. Chinda

    Mr. J.C. Nijhawan

    Ms. Rashmi Kathuria

    Ms. Divya Chetal

    Ms. Deepa Gupta

    Ms. Sharmila Bakshi

    Ms. Srelekha Mukherjee

    Science :

    Economics:

    Ms. Charu Maini

    Ms. S. Anjum

    Ms. Meenambika Menon

    Ms. Novita Chopra

    Ms. Neeta Rastogi

    Ms. Pooja Sareen

    Ms. Mridula Pant

    Mr. Pankaj Bhanwani

    Ms. Ambica Gulati

    History :

    Ms. Jayshree Srivastava

    Ms. M. Bose

    Ms. A. Venkatachalam

    Ms. Smita Bhattacharya

    Material Production Groups: Classes IX-X

    English :

    Ms. Rachna Pandit

    Ms. Neha Sharma

    Ms. Sonia Jain

    Ms. Dipinder Kaur

    Ms. Sarita Ahuja

    Science :

    Dr. Meena Dhami

    Mr. Saroj Kumar

    Ms. Rashmi Ramsinghaney

    Ms. Seema kapoor

    Ms. Priyanka Sen

    Dr. Kavita Khanna

    Ms. Keya Gupta

    Mathematics :

    Political Science:

    Ms. Seema Rawat

    Ms. N. Vidya

    Ms. Mamta Goyal

    Ms. Chhavi Raheja

    Ms. Kanu Chopra

    Ms. Shilpi Anand

    Geography:

    History :

    Ms. Suparna Sharma

    Ms. Leela Grewal

    Ms. Leeza Dutta

    Ms. Kalpana Pant

    Material Production Groups: Classes VI-VIII

    Advisory Conceptual Framework

    Ideators

    Shri Vineet Joshi, Chairman, CBSE Shri G. Balasubramanian, Former Director (Acad), CBSE

    Shri Shashi Bhushan, Director(Academic), CBSE Ms. Abha Adams, Consultant, Step-by-Step School, Noida

    Dr. Sadhana Parashar, Head (I & R),CBSE

    Ms. Aditi Misra Ms. Anuradha Sen Ms. Jaishree Srivastava Dr. Rajesh Hassija

    Ms. Amita Mishra Ms. Archana Sagar Dr. Kamla Menon Ms. Rupa Chakravarty

    Ms. Anita Sharma Ms. Geeta Varshney Dr. Meena Dhami Ms. Sarita Manuja

    Ms. Anita Makkar Ms. Guneet Ohri Ms. Neelima Sharma Ms. Seema Rawat

    Dr. Anju Srivastava Dr. Indu Khetrapal Dr. N. K. Sehgal Dr. Uma Chaudhry

    Coordinators:

    Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi, Head (I and R) E O (Com) E O (Maths) E O (Science)

    Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO

    Ms. Seema Lakra, S O Ms. Preeti Hans, Proof Reader

    Material Production Group: Classes I-V

    Dr. Indu Khetarpal Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Nandita Mathur

    Ms. Vandana Kumar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Seema Chowdhary

    Ms. Anju Chauhan Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Ruba Chakarvarty

    Ms. Deepti Verma Ms. Seema Choudhary Mr. Bijo Thomas Ms. Mahua Bhattacharya

    Ms. Ritu Batra Ms. Kalyani Voleti

  • ContentSContentS1. Syllabus

    2. Scope document

    3. Teachers' Support Material

    Teacher's Note

    Activity - Skill Matrix

    Warm up Activities

    ? W1: Activity 1

    ? W2: Activity 1

    ? W3: Activity 1

    Pre content Activities

    ? P1: Activity 1

    ? P2: Activity 1

    Graphical Representation of Pair of Linear Equations in Two Variables

    Activity 1

    Solution of Pair of Linear Equations in Two Variables Graphically

    Activity 2

    Relation between Coeff of x and y

    Activity 1

    Condition for Consistent and Inconsistent Systems

    Activity 1

    Nature of System of Linear Equations

    Activity 1

    Activity 2

    Activity 3

    Activity 4

    Solution by Substitution Method

    Activity 1

    Solution by Elimination Method

    Activity 1

    Word Search

    Activity 1

    Word Problems

    Activity 1

  • Solution by Cross Multiplication Method

    Activity 1

    Post Content PCW1

    Activity 1

    Post Content PCW2

    Activity 1

    Post Content PCW3

    Activity 1

    Post Content PCW4

    Activity 1

    4. Assessment Guidance Plan

    5. Study Materials

    6. Students' Support Material

    Worksheets and Self-Assessment Rubrics for

    Warm-up W1: Locating a point

    Warm-up W2: Separating the Linear Equation in One Variable and in two Variable

    Warm-up W3: Plotting a point

    Pre-content P1: Basic terms in Linear Equations

    Pre-content P2: Solutions of Linear Equations

    Content CW1 : Graphical representation of pair of Linear Equation in two variables.

    Content CW2 : Solution of pair of Linear Equations in two variables graphically

    Content CW3 : Relation between coff. of x and y and the number of solution for a pair of Linear Equations

    Content CW4 : Conditions for Consistent and Inconsistent systems.

    Content CW5 : Nature of System of Linear Equations

    Content CW6 : Solution by Substitution Method

    Content CW7 : Solution by Elimination Method

    Content CW8 : Word Search

    Content CW9 : Word Problems

  • Content CW10 : Solution by Cross-multiplication Method

    Post - Content PCW1 : Dialogue Dilemma

    Post - Content PCW2 : Predicting the number of solutions of equations

    Post - Content PCW3 : Cartoon based assignment on Linear Equations

    Post - Content PCW4 : Assignments based on different method of solving the pair of Linear Equations

    Post - Content PCW5 : Assignment based on Word Problems

    Post - Content PCW6 : ICT based projects

    Suggested Videos and Extra Readings

  • 1

    SyllabusUnit3:

    PairofLinearEquationsintwovariables(Core)

    GraphicalrepresentationoflinearEquations

    Plotting the lines representing two linear equations on thesameplane

    Algebraic interpretationofgraphsofsimultaneousequationsasfollowing:

    a) Intersecting lines with common point means linearequationwithuniquesolution

    b) parallel lines with no common point means linearequationswithnosolution

    c) coinciding lines with all point common means linearequationswithinfinitesolutions

    Define:consistentsystem&inconsistentsystem

    Natureofsystemoflinearequations

    Relationbetween thecoefficientsofpairof linearequationstopredictaboutthegivensystemoflinearequations

    Algebraicmethodofsolvingsystemoflinearequations

    Substitution method, elimination method and crossmultiplicationmethod.

    Applicationindailylifeproblems

    Number problems, age problems, work ratio problems,dimensionalproblems.

  • 2

    SSccooppeeDDooccuummeennttKeyterms

    1. Pairoflinearequations2. Graphicalrepresentationofpairoflinearequations3. Algebraic interpretation of Graphical representation of pair of linear

    equations4. Natureofsystemoflinearequations5. Consistentsystemofpairoflinearequations6. Inconsistentsystemofpairoflinearequations7. Substitutionmethodofsolvingpairoflinearequations8. Crossmultiplicationmethodofsolvingpairoflinearequations9. Eliminationmethodofsolvingpairoflinearequations

    LearningObjective:

    Plot the lines representing the linear equations of given system on sameplane.

    Observethatintersectinglineshaveonecommonpoint,coincidinglineshaveallpointscommonandparallellineshavenocommonpoint

    Understand the Algebraic interpretation of graphical representation i. e.intersecting lines implies unique solution ,coinciding lines implies infinitesolutionandparallellinesimpliesnosolution

    Observe the coefficients of system of linear equations a1x+b1y+c1=0 &a2x+b2y+c2=0andtheirgraphsandestablishthefollowingrelation:

    1. foruniquesolution

    2.

    forinfinitesolution

    3.

    fornosolution

    Grasp the terms consistent system and inconsistent system of linearequations

  • 3

    Predict the nature of the system of linear equations looking at thecoefficients

    Solvingthepairoflinearequationsbysubstitutionmethod Solvingthepairoflinearequationsbyeliminationmethod Solvingthepairoflinearequationsbycrossmultiplicationmethod Solvingwordproblemsbasedonthereallifesituation

    Extensionactivities:

    o UseGeoGebraandexplorewhatkindof linesareobtainedforthepairoflinearequationsax+by+c=0,dx+ey+f=0

    o when

    o when

    ,

    o when

    ,

    Lifeskill

    Check the meaning of the word consistent and inconsistent from thedictionaryandfindoutsuccessstoriesofconsistentperformers.Discusstheimportanceofbeingconsistentperformerinlife.

    Perspective

    Whatkindofgraphwillbeobtainedforlinearequationinthreevariables?How the solution will look graphically? Get some hint from followingpicturetakenfromWikipedia.

  • 4

    SEWA

    Everydayvariousschemesorplansarelaunchedbymobilecompanies.Useyour knowledge of framing and solving pair of linear equations andcomparetheplans.Educateyourfriendsandparentsaboutmakingchoicesbeforeavailinganyplan.Thisknowledgecanbeappliedtocompareallkindofschemeswhetheritisregarding insurance companiesor takinghome loanorpurchasing car ininstalments.

  • 5

    TTeeaacchheerrss

    SSuuppppoorrtt

    MMaatteerriiaall

  • 6

    TEACHERSNOTE

    The teaching ofMathematics should enhance the childs resources to think andreason,tovisualiseandhandleabstractions,toformulateandsolveproblems.AsperNCF2005,thevisionforschoolMathematicsinclude:

    1. Childrenlearntoenjoymathematicsratherthanfearit.2. Childrenseemathematicsassomethingtotalkabout,tocommunicatethrough,to

    discussamongthemselves,toworktogetheron.3. Childrenposeandsolvemeaningfulproblems.4. Childrenuseabstractionstoperceiverelationships,toseestructures,toreasonout

    things,toarguethetruthorfalsityofstatements.5. Children understand the basic structure of Mathematics: Arithmetic, algebra,

    geometryandtrigonometry,thebasiccontentareasofschoolMathematics,allofferamethodologyforabstraction,structurationandgeneralisation.

    6. Teachers engage every child in classwith the conviction that everyone can learnmathematics.

    Students shouldbeencouraged to solveproblems throughdifferentmethods likeabstraction, quantification, analogy, case analysis, reduction to simpler situations, evenguessandverifyexercisesduringdifferentstagesofschool. Thiswillenrich thestudentsandhelpthemtounderstandthataproblemcanbeapproachedbyavarietyofmethodsforsolvingit.Schoolmathematicsshouldalsoplayanimportantroleindevelopingtheusefulskillofestimationofquantitiesandapproximatingsolutions.DevelopmentofvisualisationandrepresentationsskillsshouldbeintegraltoMathematicsteaching.Thereisalsoaneedtomake connectionsbetweenMathematicsandother subjectsof study. When childrenlearntodrawagraph,theyshouldbeencouragedtoperceivethe importanceofgraph intheteachingofScience,SocialScienceandotherareasofstudy.Mathematicsshouldhelpindevelopingthereasoningskillsofstudents.Proofisaprocesswhichencouragessystematicwayofargumentation.Theaimshouldbetodeveloparguments,toevaluatearguments,tomakeconjuncturesandunderstandthattherearevariousmethodsofreasoning.Studentsshould bemade to understand that mathematical communication is precise, employsunambiguoususeoflanguageandrigourinformulation.Childrenshouldbeencouragedtoappreciateitssignificance.

    AtthesecondarystagestudentsbegintoperceivethestructureofMathematicsasadiscipline. By this stage they should become familiar with the characteristics ofMathematicalcommunications,varioustermsandconcepts,theuseofsymbols,precisionof languageandsystematicarguments inprovingtheproposition. Atthisstageastudentshouldbeableto integratethemanyconceptsandskillsthathe/shehas learnt insolvingproblems.

  • 7

    The unit on Linear Equations in Two Variables focuses on lots of geogebraicactivitiesandexplorationinordertomeetoutthefollowinglearningobjectives:

    Plotthelinesrepresentingthelinearequationsofgivensystemonsameplane.

    Observethatintersectinglineshaveonecommonpoint,coincidinglineshaveallpointscommonandparallellineshavenocommonpoint.

    UnderstandtheAlgebraicinterpretationofgraphicalrepresentationi.e.intersectinglinesimpliesuniquesolution,coincidinglinesimpliesinfinitesolutionandparallellinesimpliesnosolution.

    Observethecoefficientsofsystemoflinearequationsa1x+b1y+c1=0&a2x+b2y+c2=0andtheirgraphsandestablishthefollowingrelation:

    4.

    foruniquesolution

    5.

    forinfinitesolution

    6.

    fornosolution

    Graspthetermsconsistentsystemandinconsistentsystemoflinearequations.

    Predictthenatureofthesystemoflinearequationslookingatthecoefficients.

    Solvingapairoflinearequationsbysubstitutionmethod. Solvingapairoflinearequationsbyeliminationmethod. Solvingapairoflinearequationsbycrossmultiplicationmethod. Solvingwordproblemsbasedonreallifesituation.

    All the tasksdesigned in this chapterhavebeenprepared keeping inmind thefollowingpedagogicalissues:

    To create supportive classroom environment inwhich learners can thinktogether, learn together, participate in the discussions and can takeintellectualdecisions.

    Toprovide enough opportunities to each learnerof expression so thatteachercanhaveinsightintotheknowledgeacquired,knowledgerequired,

  • 8

    refinement required in theknowledgegainedand the thinkingprocessofthelearner.

    Emphasisoncreatingagoodcommunicativeenvironmentintheclass. Tocatervariouslearningstyles.

    Pair of linear equations in two variables is a unit in which both algebra andcoordinategeometrysharesequalrole.Usingalgebra,areal lifeproblemcanbetranslated into abstract expression involving coefficients and variables. Thealgebraic problems can be solved either using coordinate geometry (graph oflinearequations)orthroughpurealgebraicapproach.

    The study of this unit requires a clear understanding of basic concepts ofcoordinate geometry i.e. ordered pair, Cartesian plane, Cartesian coordinates,abscissa, coordinate axis, skill of drawing lines in Cartesian plane, etc. and thebasicconceptsofalgebra likealgebraicexpressions,variables,coefficients,linearequations in one variable, solution of linear equation, linear equations in twovariables,etc.

    Toattainthefirstlearningobjectivevariouswarmupactivitiescanbeusedratherthangiving instructions todirectlyplot the lines. In thewarmupactivity (W1)somepaperslipswithorderedpairwrittenover itwillbekeptonhold.Studentscanpickuptheslipsandspeakanythingwhichcomestotheirmindrelatedtothelocationofgivenorderedpair.The ideabehindthisexercise isto involveallthelearners.Noprescribedformatisgiventodescribetheorderedpair.Learnercangive any appropriate statement. Teacher can also ask the students tosimultaneously point out the quadrant on the Cartesian plane drawn onblackboard/screen. This exercise will help all the learners to brush up theirknowledgeof locatingthepointonplane.Ifany learner is lacking intheconceptthenwiththerepeatofsameexercise,everychildwillbeabletolocatethepointintheplane.Afterthisexerciseaselfassessmentsheetwithfourparameterswillbedistributedtoevery learner.Withthehelpofthisselfassessmentrubrictheycan make out whether they can identify the points located on Xaxis/Yaxis/origin/anyofthefourquadrants.Inthesamewaywarmupexercise(W3)willhelpthestudentstoplotthegivenorderedpairsonthesameplanefollowedby

  • 9

    the self assessment rubric. Warm up activity (W2) aims at brushing up theconcept of algebraic expressions, linear equations, coefficients and variables.During this activity teacher will encourage the students to speak about thewritten algebraic statement on the board/screen. Teacher can motivate thestudents to frame some real life situations where the written equation orexpressioncanmakesense.Thiswaythestudentswillbeabletorelateabstractmathematicswithdaytodayacquaintanceandwillbemotivatedto learnmoreaboutthisunit.Moreover,forthesamealgebraicexpression,differentsituationscan be quoted. This will help the group to widen their horizon and theirperspectivetoassimilatetheideas.Somestudentsalwayshesitatetosharetheirthoughtsdue to lackof confidenceordue to the feeling thatone rightanswerbeengivenandnootherrightanswerispossible.Theabovesituationcanbeusedto remove the blockage from the mind of the students and to help themunderstandthattherecouldbeseveralrightanswersorvariouspossiblesolutionstothesameproblem.Thisway,notonlythethinkingskillsofthestudentswouldbedeveloped, learnerswillalsotendtododivergentthinking.Precontenttasks(P1)and(P2)alsocoversuptheprerequisitesforplottingthe linesonCartesianplane.Before takingup theContentworksheet (CW1) in theclass, teachercanconductadiscussion in theclass regarding thenumberofpoints lyingona lineand the number of solutions which are possible for a linear equation. Brainstormingquestionscanbeaskedintheclassafterthestudentshaveplottedalineforagivenlinearequation.

    Letthestudentsobservethegraphandanswerthefollowingquestions.

    1. Howmanypointscanbelocatedonaline?2. Identifyanyonesolutionofthegivenlinearequation.3. Howmanymoresolutionsyoucanfindout?4. Is thereany correspondencebetween thepoints locatedon the lineand

    thesolutionsofthelinearequations?

    Students can gradually explore, what will happen if two linear equations areplotted on the same Cartesian plane. Teacher can give different pair of linearequationsandaskthestudentstoplotthelinesonthesameplane.Furtherthey

  • 10

    canbemotivatedtoobservethevariouspossiblerelationshipsbetweenthelines.Throughdiscussion it should comeout from students that thereareonly threepossibilitiesbetweentwogivenlines:

    1.Twolinescanbeintersecting,or

    2.Twolinescanbeparallel,or

    3.Twolinescanbecoinciding.

    Discussion can be conducted to make students think the possible values ofvariables which can satisfy both the given equations simultaneously. Let thestudentsexplore the relationshipbetween thesepossiblevalues/solutionof theequations and the lines representing them on the Cartesian plane. Once thestudents have understood that intersecting lines means unique solution,coincidinglinesmeansinfinitesolutionsandparallellinesmeansnosolution,newtermsconsistentsystemandinconsistentsystemcanbeintroduced.Apartfromthe vocabulary development, the words can be used to inculcate life skillsattitudesregardingworkcultureamongstthestudents.Ahealthydebatecanbeconducted intheclassregardingtheadvantagesofbeingaconsistentworkerordisadvantagesofbeingan inconsistentworker.Adiscussioncanalsobeheld tosensitizethestudentshowtheCCEsystemintroducedintheschoolsispreparingthe child to be a consistent performer. Consistent performance is an essentialrequisiteofmoderndayindustry.Sothestudentsneedtodevelopinthemselvestheattitudeofworking consistentlyand toavoid fluctuations inmoodatworkplace.

    After attempting a number of problems and using geogebra students can betriggered to thinkbeyond and canbemotivated to find the areaof a triangleformedbyanaxisandapairoflines.Theycanalsotryto

    1. Findouthowmanytrianglesarepossiblebytwolinesandanaxis?2. Whatarethecoordinatesofthepointswhenthetwo linesmeetatXaxis

    orYaxis?

  • 11

    Abrain storming session canbe conducted to improve the skillsof finding thesolutionofapairof linearequations in twovariableswhensomevaluesofonevariableareunknowncorrespondingtotheknownvaluesoftheothervariables.

    Tostrengthentheconceptsandtohelpthelearnerstolearnattheirownpacealotof videosavailableon youtube canbe suggestedasmentioned in teacherssupport material (activity 6 content (CW1)). Extra reading material is alsosuggestedtoestablishthemasindependentlearners.

    Further, students should be motivated to see the relationship between thecoefficients of these equations. Through numerous examples allow them toinvestigatetherelationshipbetweentheratiosofcoefficientsoftwovariablesxandyaswellasconstantterms.Studentsshouldbeabletoreachtheconclusionthatforagivenpairofequationsa1x+b1y+c1=0anda2x+b2y+c2=0

    Conditions:

    1. foruniquesolution

    2.

    forinfinitesolution

    3.

    fornosolution

    Usinggeogebraactivitiesthelearningcanbeextendedtoexploretherelationbetweenthelineswhen

    1.

    2.

    3.

    Byobservingtheratioofthecoefficientsstudentscanmakeoutthatwhensystemofequationshasinfinitesolutionstheequationsaredependentoneachother.So,newtermslikeindependentsystemanddependentsystemcanbeintroduced.Withalltheaboveexercises,althoughselfassessmentrubricforthestudentsaresuggestedattheendofeachworksheet,toreinforceandto

  • 12

    strengthentheconcepts,contentworksheet(CW4)and(CW5)focusonassimilationofalltheseconcepts.

    Beautyofmathematics lies in the factthattherearevariouspathstoreachthesameconclusion.Same istrueforfindingthesolutionof linearequations.Linearequations can be solved algebraically too without using any graph. Variousmethodslikesubstitution,elimination,crossmultiplicationmethodcanbeusedtodoso.Eachmethodcanbeexplainedtothestudentswiththehelpofexamplesand can be mastered by them by attempting a good number of problems.Studentscanalsobemotivatedtosolvethegeneralequationsa1x+b1y+c1=0anda2x+b2y+c2=0andtogeneralizethesolutionsasformulae.

    To revise all the concepts and to strengthen the vocabulary related to thechapter,word search in the form of contentworksheet (CW 8) is given. Thisactivityisframedinordertobreakthemonotonyofdrillingtheproblemsolve.Aninteresting activity, dialogue dilemmawith, the help of cartoon strips, can beconducted in the class.After reading thedialoguesbetween various charactersillustrated in cartoon strips students will give answers based on all conceptslearntinthechapter.

    Toachievethelastlearningobjectivei.e.tosolvereallifeproblemswiththehelpof linearequations,studentscanbeaskedtoposesomesituationswhichcanbetranslatedintolinearequations.Theycanbegivencertainsituationsandaskedtoconvertthemintolinearequationsandgraduallycanbemotivatedtosolvethemusing any of the above learnt methods. Throughout the development of thechapter teacher should keep in mind not to introduce any term or conceptdirectly. Focus should be on taking the students on a journey of explorationwhere they can use their previous knowledge and identify the new relations.Althoughgeogebra software is recommended forallexplorationsbut incaseofunavailability it should not be taken as drawback. Rather student should beencouraged to draw the graphs, sit in groups and explore the desired results.Teachers are also suggested not to discourage any child for asking unusualquestionorgivingunusualresponse.Somestudentscanaskwhatwillhappen ifthe linearequationscontainsmorethantwovariables.Theycanbeencouraged

  • 13

    to findmoreabout linearequations in threeormorevariableswith thehelpofgeogebra or the information available on web. Teacher should divert theresponseinrightdirectionbyaskingappropriatequestions.

    Attheendstudentsshouldalsobeencouragedto identifytheareaswheretheycanusetheknowledgeacquiredinthischapter.Forexample,theadvertisementsgivenbyvariousagencies innewspapersandmagazines toavail thecar loanorthehouse loanetc.canbecomparedusingknowledgeofsolvingapairof linearequations.Overall theentireunit canbeused toenhanceexploratory skills, toenhancedivergentkindofthinking,tofindtherelationbetweentheabstractandpractical situations and to nurture creativity alongwith inculcating the valuesregardingworkculturerequiredfor21stcentury.

  • 14

    ActivitySkillMatrix

    TypeofActivity NameofActivity SkilltobedevelopedWarmUP(W1) Locatingapoint Spatialskill,geometricalskill,drawingof

    graphWarmUP(W2) Seperatingthe

    linearequationsinonevariableandintwovariables

    knowledgeandunderstanding

    WarmUP(W3) Plottingapoint Geometricalskill,drawingofgraphPreContent(P1) Basictermsin

    linearequationKnowledgeandunderstanding

    PreContent(P2) Solutionoflinearequation

    Synthesisofinformation

    Content(CW1) Graphicalrepresentationofpairoflinearequationintwovariables

    Spatialskill,geometricalskill,drawingofgraph

    Content(CW2) Solutionofpairoflinearequationsintwovariablesgraphically

    Drawingandinterpretationofgraph

    Content(CW3) Relationbetweencoefficientofxandyandnumberofsolutionsforapairoflinearequation

    Applicationandcomputationalskills

    Content(CW4) Conditionsforconsistentandinconsistentsystems

    Memory,knowledge,understandinganapplication

    Content(CW5) Natureof Thinkingskills

  • 15

    systemsoflinearequations

    Content(CW6) Substitutionmethod

    Computationalskills.

    Content(CW7) EliminationMethod

    Computationalskills.

    Content(CW8) Wordsearch MemoryandobservationContent(CW9) Wordproblems

    Problemsolvingskills,Thinkingskillandcomputationalskills

    Content(CW10) Crossmultiplicationmethod

    Computationalskills

    PostContent(PCW1)

    DialogueDilemma

    Analyticalskills,communicationskills,thinkingskills.

    PostContent(PCW2)

    Predictingthenumberofsolutionofequations

    Computationalskills.

    PostContent(PCW3)

    Cartoonbasedassignmentonlinearequations

    Observation,expression,knowledgeandcomprehension

    PostContent(PCW4)

    Assignmentbasedondifferentmethodsofsolvingthepairoflinearequations

    Problemsolving,computationalskills

    PostContent(PCW5)

    Assignmentbasedonwordproblems

    Problemsolving,computationalskills

    PostContent(PCW6)

    ICTbasedproject eskills,observationskills,Analyticalskillsandsyntheticskill,organizingandcompilingofdata

  • 16

    Activity1warmup(W1)

    Specificobjective:

    TorecallthelocationoftheorderedpairsintheCartesianplane

    Description: In the earlier classes the students gained the knowledge ofquadrants in the Cartesian plane. This is a starter activity through which thestudents will bemotivated for learning the plotting of ordered pairs in theirrespectivequadrants.Eachstudentwillspeakabouttherespectiveorderedpair,itslocationinthequadrant/xaxis/yaxis/origin.

    Execution: Takesomepaperslipsandwriteanorderedpaironeach.Putallofthem inabowl.Eachstudentwillpickuponepaperslipandtellthe locationoforderedpairsintheCartesianplane.

    Note:LaystressondifferentlocationsintheCartesianplane.Theorderedpair(0,0)isorigin.Apointonthexaxisisoftype(x,0)andontheyaxisisofthetype(0,y).Talkaboutthesignsofabscissaandordinatesinfourquadrants.

    Parametersforassessment:

    Studentsareabletotell

    Locationofapointinfourquadrants Locationofapointonthexaxis Locationofapointontheyaxis LocationoftheOrigin.

    Extrareading:http://www.teacherschoice.com.au/maths_library/coordinates/plotting_ordered_pairs.htm

    http://www.webmath.com/gpoints.html

  • 17

    Video Watch: You may ask the students to watch videohttp://www.youtube.com/watch?v=Hut9QnQlF8&NR=1&feature=fvwp forrevisionofplottingofpointsinaCartesianplane.

    Activity2warmup(W2)

    Specificobjective:Torecalllinearequationinonevariableortwovariables.

    Description: Inearlier classes the students learntof thedifferencebetweenanalgebraic expression and an equation. Through thiswarm up activity theywillbrushup their knowledge by recognizing the algebraic expressions and linearequations.

    Execution:Teacherwillwritesomealgebraicexpressions,linearequationsinonevariable and two variables. Studentswill speak outwhether it is an algebraicexpressionorlinearequationinonevariableortwovariables.

    Note:Tomake the taskmore interesting youmayprepare flash cardsandusethemintheclass.

    Parametersforassessment:

    Studentsareableto

    Identifyalinearequationinonevariable. Identifyalinearequationintwovariables.

  • 18

    Activity3warmup(W3)

    Specific objective: To recall the knowledge of Cartesian plane and locate theorderedpairsshownontheCartesianplane.

    Description: Intheearlierclassesthestudentshavegainedtheknowledgeofallthefourquadrantsformedbythetwoaxes.

    Theywillusetheirknowledgeandtheconceptofquadrantstolocatetheorderedpairsintheirrespectivequadrant.

    Execution:Distributetheworksheetsandaskthestudentsto locatetheorderedpairsonthegraph.

    Parametersforassessment:

    Locationofapointinaquadrant Locationofapointonthexaxis Locationofapointontheyaxis LocationofOrigin.

    Extrareading:http://www.mathsteacher.com.au/year8/ch15_graphs/01_cartesian/plane.htm

  • 19

    Activity4Precontent(P1)

    Specificobjective:Tomake thestudents recall the termscoefficients,variables,constanttermofalinearequationintwovariables

    Description: This is a pre content activity for refreshing students knowledgeaboutvarioustermsrelatedtoalinearequationintwovariables.Usingthegivencoefficientsand thegiven statements studentswillconvert the statements intoMathematical expressions. Students can also be encouraged to narrate somesituationscorrespondingtotheobtained linearequation.Teachermusthelpthestudents to realize that to such kind of questionsmore than one answers arepossible.

    Execution:Teachermayaskthestudentstowritealinearequationaccordingtothe given condition. There can be varied answers. Let students speak theiranswersanddiscussthem.

    Parametersforassessment

    Studenthastheknowledgeof

    Termslikecoefficients,variables,constantterm,linearequation

    Extrareading:

    http://www.mbs.edu/home/jgans/mecon/value/Popups/pop_up_analytical_methods.htm

  • 20

    Activity5Precontent(P2)

    Specificobjective:Tomakethestudentsrecallwhatasolutionofanequationis?

    Description:Inclass9studentshavelearnttofindsolutionsofalinearequationinonevariableaswellaslinearequationintwovariables.Theywillbemotivatedpredictthesolutionofalinearequationintwovariables.Eachstudentwillcheckifthegivenorderedpairsatisfiesthegivenlinearequationintwovariables.

    Execution:Afterdistributingtheworksheets,studentswillbeaskedtosubstitutetheabscissaandordinateoftheorderedpairinthegivenequationandcheckifitsatisfiestheequation.Ifitsatisfiesthenitisasolutionotherwisenot.

    Pointstonote:Therecanbemorethanonesolutionofanequation.Allthepointswhichare solutionof the givenequationwill lieon the graphof theequation.Discussioncanbeheldtomakethelearnersrealizethatalinearequation intwovariablescanhave infinitesolutions.Eachsolutionwillcorrespondtoapointonthelinerepresentinglinearequationintwovariables.

    Parametersforassessment

    Abletofindsolutionofalinearequationintwovariables

    Followup:WatchVideo

    http://www.youtube.com/watch?v=cHH_NqNuwYI

  • 21

    Activity6Content(CW1)

    Specificobjective:

    Tolearntoplotapairoflinearequationsintwovariablesonthegraph

    Description: In this content activity, the students are encouraged to plot thecoordinates(orderedpairs)onthegraph,givenineachtableandtojointhemandidentifywhat theyget.Theyhave to locatemorepointson itand identifyhowmanymorepointscanbethereonit?

    DiscusstheKeyWords:Coincident,Parallel,Intersecting

    Pointstoremember:Stressonthefollowingpoints

    Labeltheaxes. Putarrowsonthelineends. Orderedpairsaremarkedwithcapitallettersandwrittenas(x,y).Herex

    isabscissaandyisordinate.

    Execution:Eachstudentplotsthecoordinatesofeachtableandjoinsthem.Theyseewhattypeoflinesdotheyget,dotheyhaveanypointcommonornot.

    Parametersforassessment:

    LocatethepointsontheCartesianplane. Abletojointhepointsandobtainthelines. Identifytypesoflinesobtained. Locatethecommonpointsonthetwolines.

    Followup:Youmayaskthestudentstowatchvideos

    http://www.youtube.com/watch?v=VKqledd8wUA

    http://www.youtube.com/watch?v=4h65y9Xj4eY&feature=related

  • 22

    Extrareading:

    Graphinglinearequationsintwovariables

    http://www.tpub.com/math1/13.htm

    http://www.purplemath.com/modules/graphlin.htm

    Video:

    http://www.youtube.com/watch?v=jpVrLZdIuW0

    http://www.mathexpression.com/graphing.html

    Brainstormingquestionsfordiscussion:

    1. The table shows four pairs of values of x and y that satisfy the linearequationy=2x3.Findthevaluesformandn.

    X 0 1 1 2Y 3 m 5 n

  • 23

    2. Thegraphshows fivepointsA,B,C,DandE.Whichofthe3pointswhenconnectedtogetherwillsatisfythelinearequationofxy=0?

    Activity7ContentWorksheet(CW2)

    Specificobjective:To learntorepresentthe linearequations intwovariablesonthe graph and read the solution of the pair of equations on the basis ofintersectinglines,coincidentlinesorparallellines.

    Description:Intheearlierclassesthestudentshavelearnedtodrawthegraphofalinearequationintwovariables.Theconceptofsolutionoflinearequationswillbebuiltupby takingmorepairof linesanddraw thegraphs.Further theywillobtainthesolution(s),ifany.

    Execution:

    Each student will be given a worksheet to find the ordered pairs whichsatisfies each equation separately. On the provided graph sheet studentswilldrawthegraphbyjoiningtheorderedpairsobtained.Thenthetypeof lineswillbe identified and the commonpoints (if any) are found.On thebasisof it thesystemwillbetermedconsistentorinconsistent.

  • 24

    Key words for discussion: Intersecting, Coincident, Parallel, Consistent andInconsistent

    Parameters:

    Findingsuitableorderedpairs LocatethepointsontheCartesianplane. Jointhepointscorrectly. Identifythetypesoflinescorrectly.

    Extrareading:

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

    http://www.purplemath.com/modules/systlin2.htm

    Brainstormingquestionsfordiscussions:

    Given below is a graph representing pair of linear equations in two variables.x+y=4,3x2y=12Observethefollowingcarefully...

  • 25

    Thegiventwo lines intersectat(4,0)which isthesolutionofgivenpairoflinearequationsintwovariables.

    CoordinatesofpointswherelinescuttheyaxisareA(0,4)andC(0,6) Verticesoftriangle formedbygiven linesandyaxisareA(0,4),B(4,0)and

    C(0,6) TheareaoftriangleABC=1/2(10x4)=20squareunits

    Given below is the graph representing pair of linear equations in twovariables

    xy=4,x2y=4

    Thegiventwo lines intersectat(4,0)which isthesolutionofgivenpairoflinearequationsintwovariables.CoordinatesofpointswherelinescuttheyaxisareA(0,4)andC(0,2)VerticesoftriangleformedbygivenlinesandyaxisareA(0,4),B(4,0)andC(0,2)TheareaoftriangleABC=1/2(2x4)=4squareunits

    Given below is a graph representing pair of linear equations in two variablesxy=2,x+y=4Answerthefollowingquestions:

  • 26

    1. Whatarethecoordinatesofpointswheretwolinesmeetxaxis?2. Whatarethecoordinatesofpointswheretwolinesmeetyaxis?3. Whatisthesolutionofgivenpairofequations?Readfromgraph.4. Whatistheareaoftriangleformedbygivenlinesandxaxis?5. Whatistheareaoftriangleformedbygivenlinesandyaxis?

  • 27

    ExtraPracticeWorksheet

    1.

    x - y

    16

    =

    1

    4

    x - y = 4

    2. 5 +

    2

    5y = x

    5x - 4y = 25

    3.

    5x - 4y = -5 10y = 74 - 8x

    4.

    8=

    -4

    3

    y + 4x

    y =

    1

    4

    x -

    13

    4

    5.

    y = x + 1 -4x = 8 - 2y

    6.

    y = 4x + 6

    y = 4

    3

    x +

    2

    3

    Createdusinghttp://www.edhelper.com/LinearEquations.htm

  • 28

    Activity8ContentWorksheet(CW3)

    Specificobjective:

    Thestudentswillbeabletoidentifythecoefficientsofthevariablesandconstanttermsinthegivenpairofequations.

    Theywillbeabletofindarelationbetweentheratiosofthecorrespondingcoefficientsofthetwovariablesandtheconstantterms

    Description:

    Thestudentsarealreadyawareofthetermcoefficientofvariableandconstantterms,theywillfurtheridentifythecoefficientofthevariablesinbothequationsandthenfindtheirratioandfindarelationbetweentheseratioswhenlineshaveuniquesolution,nosolutionandmanysolutions.Theywillalsoinvestigatethetypesofthegraphandtherelationbetweentheratiosofthecoefficientsandconstantterms.

    Execution:

    Eachstudentwillidentifythecoefficientsofthevariablesandtheconstanttermsandfindtheratioofcoefficientsofcorrespondingvariablesintwoequations.Studentswillbeaskedtoobservethegivengraphofapairoflinearequationandcomparetheratiosofcoefficientsofx,ratioofcoefficientofyandconstantterms.Fromthegraphstheywillinvestigatetheconditionsforconsistencyandinconsistencyforgivensystem.

    Parameters:

    Identifythecoefficientsofthevariables. Identifytheconstantterms

    Calculatetheratio , , Identifythetypeofpairoflinearequations. identifythenumberofsolution(s)

  • 29

    Activity9ContentWorksheet(CW4)

    Specificobjective:Tolearnaboutconsistentandinconsistentpairofequations

    Description:Itisanactivityfollowedbythelearningofvarioustypesofgraphicalrepresentationsofpairofequationsandvarioustypesofsolutions.

    Execution:Askthestudentstorecallthetypesofgraphsobtainedviz.intersecting,parallelandcoincidentlines.Therearetwocaseswhenthereisatleastonesolution.Letstudentsdiscussallthecasesandthefactthatwhenweareabletofindtwosolutionsforagivenpairofequationsthenwewillhaveinfinitesolutions.Insuchacasewewillgetcoincidentlines.

    Afterexplainingthetermsconsistentandinconsistentsystem,studentswillbegivenContentworksheet(CW4).Studentswillsolvethequestionsandafterwardstherewillbeadiscussionontheanswers.

    Parameters:

    Explanationoftwotypesofsystemsviz.consistentandinconsistent Writingpairofconsistentequations Writingpairofinconsistentequations

    Extrareading:

    Followup:VideoWatchhttp://www.youtube.com/watch?v=R4FkXOgDQc

    http://www.youtube.com/watch?v=VqWfxtc2vCg&feature=related

  • 30

    Activity10ContentWorksheet(CW5)

    Specificobjective:Tobeabletoidentifytherelationshipbetweentheratioofthecoefficientsandconstantwithtypeoflines.

    Description: In earlier classes the students have already learnt the relationbetween the ratio of coefficients and constantswith the type of pair of linearequationsintwovariables.InCW5theyarefurthergivensituationsinwhichtheywillbeasked to formequationswhichwillbe consistentor inconsistent to thegivenequation.

    Execution:Recall, thepreviousclassknowledgeandencourageeach student toformulate another equation or completes the equations with constantsdependinguponthetypeoflinesortypeofsystem(consistentorinconsistent)

    Parameters:

    Writingconsistentpairsofequations Writinginconsistentpairsofequations

  • 31

    Activity11ContentWorksheet(CW6)

    Specificobjective: To learntosolvepairof linearequationsalgebraicallyusingsubstitutionmethod.

    Description: InCW 6 the teacherwill ask the students to expressone variablefromanyoneequationintermsoftheother.Manyexampleswillbediscussedtogrilltheconcept.Furtheritwillbeexplainedtostudentsthatifthatexpressionissubstituted in theother equation then itbecomes an equation inone variablewhichcanbesolvedeasily.

    Execution:

    Each studentwill express one variable in terms of the other form firstequationandsubstituteitsvalueintheothertosolvethesecondequation,whichbecomesanequation inonevariable.Thesolutionof itwillgivethestudentthevalueofthefirstvariable.

    Parameters:

    Expressingonevariableintermsofother. Substitutingforexpressionintheotherequation. Solvingthesecondequation. Gettingthevalueoffirstvariable. Findingthesolution(ifany)

    Extrareading:

    http://www.mathguide.com/lessons/Systems.html

    Followup:VideoWatchhttp://www.youtube.com/watch?v=cwHR_B9zK7k&feature=related

  • 32

    Activity12ContentWorksheet(CW7)

    Specificobjective:

    Studentswillbeable tosolvepairof linearequationsalgebraicallyusingeliminationmethod.

    Description:

    Studentsareawareofsolvingpairof linearequations in twovariablesbysubstitution method. Moving ahead teacher will ask students to identify thecoefficientsof thevariables.The teacherwillexplain thatwecaneliminateonevariable if the coefficients of the variable are same in both equations, bysubtracting the equations. In thismanner, one variablewill be eliminated andvalueofothercanbecalculated.

    Execution:

    Students will make the coefficients of either of the variables same bymultiplyingthewholeequationbyacertainconstant(sothattheresultisLCMofthecoefficients).Thenthestudentswillsubtracttheequationsandeliminateonevariable and find the value of the other. By substituting its value in any oneequationthestudentwillobtaintheeliminatedvariablesvalue.

    Parameters:

    Makecoefficientofonevariablesame. Subtract/addtheequations. Obtainthevalueofeachvariable. Getsthevalueoffirstvariable.

    Extrareading:http://rachel5nj.tripod.com/NOTC/ssoewog2.html

    http://www.purplemath.com/modules/systlin5.htm

  • 33

    Activity13ContentWorksheet(CW8)

    Specificobjective:Torecapitulatethevarioustermslearntinthechapter.

    Description:Thisisawordsearchfunactivity.Itisatimeboundactivity.

    Execution:Eachstudentwillbegiventhepuzzletemplateandaskedtosearchthewords learnt in thechapterswhichareplacedhorizontally,diagonally,verticallyaswellasbackwards.Allthestudentswillbegiven15minutesfortheactivity.

    Parameters:

    FindingthecorrectwordsSolution

    Youmaycreateawordsearchusingtheonlinetoolhttp://www.armoredpenguin.com/wordsearch/

  • 34

    Activity14ContentWorksheet(CW9)

    Specificobjective:To learntoapplytheknowledgeofpairof linearequations insolvingproblems.

    Description:After learning tosolve thepairof linearequations in twovariablesusingdifferentmethods,applicationpartwillbeexploredthroughvarioustypesof word problems. Initially students will be asked to frame mathematicalexpressionsfromsmallstatementsandthenframeapairofequations.

    Execution:Throughdiscussingvarioustypesofwordproblems,theapplicationofpairofequationswillbeappreciated.

    Parameters:

    Abletoframemathematicalequations Abletosolvethepairofequations

    Extrareading:http://www.purplemath.com/modules/systprob.htm

  • 35

    Activity15ContentWorksheet(CW10)

    Specificobjective:Tolearntousethecrossmultiplicationmethodforsolvingthegivenpairoflinearequationsintwovariables

    Description:After learning tosolve thepairof linearequations in twovariablesusingdifferentmethods,viz.substitutionmethods,eliminationmethod,studentswill be given an exposure to solve the pair of linear equations using crossmultiplicationmethod.

    Execution:Explainthemethodanddiscuss.Distributetheworksheettostudentsto explore the method by solving some problems using cross multiplicationmethod.

    Parameters:

    Abletosolvetheproblemsusingcrossmultiplicationmethod

    PostContentActivities

    Activity16PostContentWorksheet(PCW1)

    Activity17PostContentWorksheet(PCW2)

    Activity18PostContentWorksheet(PCW3)

    Activity19PostContentWorksheet(PCW4)

    Activity20PostContentWorksheet(PCW5)

    Activity21PostContentWorksheet(PCW6)

  • 36

    AssessmentPlan

    Assessmentguidanceplanforteachers

    Witheachtask instudentsupportmaterialaselfassessmentrubric isattachedforstudents.Discusswiththestudentshoweachrubriccanhelpthemtokeepintune theirownprogress.These rubricsaremeant todevelop the learneras theselfmotivatedlearner.

    Toassess the studentsprogressby teacher two typesof rubricsare suggestedbelow,oneisforformativeassessmentandoneisforsummativeassessment.

    SuggestiveRubricforFormativeAssessment(exemplary)

    Parameter Mastered Developing Needsmotivation

    Needspersonalattention

    Algebraicmethodofsolvingpairoflinearequationsintwovariablesxandy

    Abletoapplysubstitutionmethodtosolvethepairoflinearequations,getcorrectvalueofxandy,canverifythecorrectnessofsolution

    Abletoapplysubstitutionmethodtosolvethepairoflinearequations,getcorrectvalueofxandy,cannotverifythecorrectnessofsolution

    Abletoapplysubstitutionmethodtosolvethepairoflinearequations,notabletogetthecorrectvalueofxandy

    NotAbletoapplysubstitutionmethodtosolvethepairoflinearequations.

    Abletoapplyeliminationmethodtosolvethepairoflinearequations,getcorrectvalueofxandy,canverifythe

    Abletoapplyeliminationmethodtosolvethepairoflinearequations,getcorrectvalueofxandy,cannot

    Abletoapplyeliminationmethodtosolvethepairoflinearequations,cannotgetcorrectvalueofxandy,

    NotAbletoapplyeliminationmethodtosolvethepairoflinearequations.

  • 37

    correctnessofsolution

    verifythecorrectnessofsolution

    cannotverifythecorrectnessofsolution

    Abletoapplycrossmultiplicationmethodtosolvethepairoflinearequations,getcorrectvalueofxandy,canverifythecorrectnessofsolution

    Abletoapplycrossmultiplicationmethodtosolvethepairoflinearequations,getcorrectvalueofxandy,cannotverifythecorrectnessofsolution

    Abletoapplycrossmultiplicationmethodtosolvethepairoflinearequations,cannotgetcorrectvalueofxandy,

    Notabletoapplycrossmultiplicationmethodtosolvethepairoflinearequations.

    Fromaboverubricitisveryclearthat

    Learner requiringpersonalattention ispoor inconceptsand requires thetrainingofbasicconceptsbeforemovingfurther.

    Learner requiring motivation has basic concepts but face problem incalculationsorinmakingdecisionaboutsuitablesubstitutionetc.Hecanbeprovidedwith remedialworksheets containing solutionmethodsof givenproblemsintheformoffillups.

    Learnerwhoisdevelopingisabletochoosesuitablemethodofsolvingtheproblemandisabletogettherequiredanswertoo.Mayhavethehabitofdoing things to the stagewhere thedesired targetscanbeachieved,butavoid going into finer details or towork further to improve the results.Lerneratthisstagemaynothaveanymathematicalproblembutmayhaveattitudinalproblem.Mathematicsteacherscanavailtheoccasiontobringpositiveattitudinalchangesinstudentspersonality.

    Learnerwhohasmasteredhasacquiredalltypesofskillsrequiredtosolvethepairoflinearequationsintwovariables.

  • 38

    TeachersRubricforSummativeAssessmentoftheUnit

    Parameter 5 4 3 2 1Solvingpairoflinearequationsgraphically

    Abletoexpressonevariableintermsofothervariable

    Abletodrawtablefordifferentvaluesofxandy

    Abletoplotallpointsoncoordinateplanaccuratelyandgetthelinesbyjoiningthem

    Abletopredictthenumberofsolutionsaccordingtolinesasuniquesolution/nosolution/infinitesolution.

    Abletoidentifythesystemasconsistentorinconsistent

    Abletoreadtheexactpointofintersectionandfindthesolution.

    Notabletoexpressonevariableintermsofothervariable

    Notabletodrawtablefordifferentvaluesofxandy

    Notabletoplotallpointsoncoordinateplanaccuratelyandgetthelinesbyjoiningthem

    Notabletopredictthenumberofsolutionsaccordingtolinesasuniquesolution/nosolution/infinitesolution.

    Notableto

  • 39

    identifythesystemasconsistentorinconsistent

    Notabletoreadtheexactpointof

    Solvingpairoflinearequationsalgebraically

    Abletoapplysubstitutionmethodcorrectlyandcanfindthecorrectsolutionandcanverifyit.

    Abletoapplyeliminationmethodcorrectlyandcanfindthecorrectsolutionandcanverifyit.

    Abletoapplycrossmultiplicationmethodcorrectlyandcanfindthecorrectsolutionandcanverifyit.

    Abletoidentifythemostsuitablemethodofsolvingthepairoflinearequationsandcanfindthecorrectsolutionandcanverifyit.

    Notabletoapplysubstitutionmethodcorrectlyandcannotfindthecorrectsolution.

    Notabletoapplyeliminationmethodcorrectlyandcannotfindthecorrectsolution.

    Notabletoapplycrossmultiplicationmethodcorrectlyandcannotfindthecorrectsolution.

    Notabletoidentifythemostsuitablemethodofsolvingthepairoflinear

  • 40

    equationsandcannotfindthecorrectsolution

    Applicationinwordproblems

    Abletoidentifythevariablesfromgivenstatement

    Abletoframepairoflinearequationscorrectly

    Abletosolvetheequationsbyanyoftheabovemethods

    Abletoverifythesolution.

    Notabletoidentifythevariablesfromthegivenstatement

    Notabletoframepairoflinearequationscorrectly

    Notabletosolvetheequationbyanyoftheabovemethods

    Notabletoverifythesolutions

  • 41

    SSTTUUDDYY

    MMAATTEERRIIAALL

  • 42

    ax+by+c=0,wherea,b,carereal

    numbersandaandbarenonzero

    iscalledalinearequationintwo

    variablesxandy.

    PAIROFLINEAREQUATIONSINTWOVARIABLES

    3.1Introduction

    Youarealreadyfamiliarwith linearequations inonevariableandtwovariables.Recalltheequationslike2x+4=0,3y5=0etc.areexamplesoflinearequationsinonevariableandequationslike2x+5y=9andy=3x+2areexamplesoflinearequationsintwovariables.

    Ingeneralanequationoftheform

    .

    Recall,asolutionofalinearequationinonevariableisarealnumberwhichwhensubstitutedforthevariablesmakestheequationtrue.

    For example solution of equation 3q 2 =0 is q = . This value of q on

    substitutionin3q2willequatetozero.

    Ingeneralanequationoftheform

    ax+c=0,wherea,carereal

    numbersandaisnonzeroiscalleda

    linearequationinonevariablex.

    (a) 2x = 1 is a linear equation in onevariablex.

    (b)3q2=0isalinearequationinonevariableq.

    (c)0.3m=2.1isalinearequationinonevariablem.

    (a) 2x 3y = 1 is a linear equation in twovariablesxandy.

    (b)5p3q 2=0 isa linearequation in twovariablespandq.

    (c)0.2+0.3m=2.1 isa linearequation intwovariablesandm.

  • 43

    Asolutionofalinearequationintwovariablesisapairofnumbers;oneforeachvariable,whichwhensubstitutedfortherespectivevariablesmakestheequationtrue.

    Forexample,2x+3y=7hassolutionx=2andy=1.

    Assubstitutionofthesevaluesofxandyin2x+3yreduceitto2(2)+3(1)=4+3=7.So,x=2andy=1isasolutionof2x+3y=7.

    Thissolutioncanalsobewrittenasanorderedpair(2,1)whichmeansx=2,y=1.

    Wordofcaution:orderedpair(2,1)(1,2).(2,1)impliesthatx=2,y=1,while(1,2)impliesx=1,y=2

    Observethegraphofequation2x+3y=7.

  • 44

    Howmanypointsyoucanfindfromgraphthatsatisfiestheequation2x+3y=7.

    Howmanysolutionofthisequationarepossible?

    Isittrueforalllinearequations?

    Notethatx=3,y= andx=3,y= arealsosolutionsoftheequation2x+3y=

    7. You are also aware of graphical representation of a linear equation in twovariables.Thegraphofa linearequation isastraight linesuch thattheorderedpairs representingpointson the lineare the solutionsof theequationand theorderedpairsnotonthelinearenotthesolutionsoftheequation.

    Asa linehasan infinitenumberofpointson it,we canagain see thata linearequationintwovariableshasaninfinitenumberofsolutions.

    Whatwill be number of solutions if there is a pair of linear equations in twovariablessayxandysatisfyingboththeequations?

    Will thenumberof solutions insuchacasebestill infiniteoruniqueor twoornone?

    Letusexamineallthesequestions.

    3.2PairofLinearEquationsinTwoVariables

    Two linearequations inthesametwovariablestakentogether issaidto formapairoflinearequations.

    Thegeneralformforapairoflinearequationsintwovariablesxandyis

    a1x+b1y+c1=0

    Pointtoremember

    Alinearequationintwovariableshasinfinitelymanysolutions.

  • 45

    a2x+b2y+c2=0

    Wherea1,b1,c1,a2,b2,c2areallrealnumbersanda12+b1

    20,a22+b2

    20.

    (i) x+y2+0 and 3x5y+2=0(ii) 2x+y=10 and 4x=3y(iii) 3x+2y=12 and 3x+2y=18(iv) 2u+3v=5 and 4u+6v=10

    Forexample,x=2andy=1i.e.(2,1)isasolutionofthepairofequation.

    2x+3y=7

    3x2y=4

    Onsubstitutingx=2andy=1,inthegivenequations,weseethatlefthandsideexpressionbecomesequal to the value givenon righthand sideof theequations:

    2x+3y=7 3x2y=4

    2.(2)+3.(1)=7 3.(2)2.(1)=4

    4+3=7 62=4

    7=7 (True) 4=4 (True)

    So,wecansaythatx=2,y=1satisfyboththeequations.

    Thus,x=2,y=1isasolutionofthegivenpairofequations.

    A solution toapairof linearequations intwo variables is an ordered pair of realnumberswhichsatisfyboththeequations.

  • 46

    Example1:

    Checkwhethertheorderedpair(2,3)isasolutionofgivenpairoflinearequationsintwovariables.

    3x2y=0

    4x3y=1

    Solution:

    Substitute2forxand3foryinboththeequations,weget

    3x2y=0 4x3y=1

    3.(2)2.(3)=0 4.(2)3.(3)=1

    66=0 89=1

    0=0 (True) 1=1(False)

    So,(2,3)doesnotsatisfytheequation4x3y=1.

    Thus,(2,3)isnotasolutionofthegivenpairoflinearequationsintwovariables.Sofar,wehavecheckedwhetheragivenorderedpairisasolutionofagivenpairof equations or not. Now our next step will be how to find an ordered pair(solution)satisfyingthegivenequations.

    Forthatpurpose,therearemanymethodsgeometric(graphical)andalgebraic.

    Think!

    Linearequationsalwaysrepresentalineincoordinateplane.Whentwolinesaredrawnonthesameplane,whatkindofpossiblerelationstheycanhave?

    Doessolutionhasanygraphicalsignificance?

  • 47

    3.3GraphicalRepresentationofAPairofLinearEquationsintwovariables

    Letusexaminetheabovestatedproblemsbydrawing graphsofdifferentpairsoflinearequations:

    Example2:

    Drawthegraphsofequations4x3y=6andx+2y=7onthesameCartesianplane.

    Note:Aslearntearlier,todrawthegraphoflinearequationsfollowthefollowingsteps:

    1. Expressoneofthevariablesintermsoftheother.2. Drawthetableofdifferentvaluesofxandy.3. Plotthepointstogetthelines.

    Solution:Tablesofvaluesfortheequations:

    4x3y=6 x+2y=7

    Or y= or,x=72y

    plotthepointsonCartesianplanetoobtainlinesl1andl2asshownbelow:

    X 0 3 3Y 2 2 6

    x 7 3 1y 0 2 3

  • 48

    ObservethattheselinesintersecteachotheratonepointP.Everypointonline1givesasolutionofequation4x3y=6andeverypointonline2givesasolutionofx+2y=7.

    Thesolutionofthepairoflinearequationsisauniquepoint(3,2)i.e.x=3,y=2.

    Thus, if the two lines representingapairof linearequations intersect, then thepointofintersectionistheuniquesolutionofthegivenpairoflinearequationsintwovariables.

  • 49

    Example3:

    Drawthegraphofthepairoflinearequations2x+3y=6and4x+6y=24.

    Solution:

    Tablesofvaluesfortheequations:

    2x+3y=6 4x+6y=24

    X 0 3 3

    Y 2 2 4

    x 0 6 3

    Y 4

    0 2

  • 50

    Theline1representsthegraphof2x+3y=6andtheline2representsthegraphof4x+6y=24.

    Observethat1and2areparallellinesi.e.theydonotintersecteachother.Thus,thereisnopointofintersectionandhencethereisnosolutionofthispairofequations.

    Example4:

    Drawthegraphsofpairoflinearequations3x2y=5and6x=4y+10.

    Solution

    Tablesofvaluesfortheequations:

    3x2y=5 6x=4y+10

    X 1 1 3Y 1 4 2

    x 1 3 1y 1 2 4

  • 51

    Diagram

    The lines1and2representingtheequations3x2y=5and6x=4y+10arecoincident,i.e.thelinesintersectatinfinitelymanypoints,implying,thereareaninfinitenumberofsolutionsforthispairoflinearequationsintwovariables.

    Fromtheexamples2to4weseethatgraphofapairoflinearequationsisapairoflineswhichmaybeintersectingormaybeparallelorcoincident.

  • 52

    3.4ConsistentandInconsistentsystemofLinearEquations

    Natureofsystemoflinearequations

    Whenapairoflinearequationsintwovariableshasoneormoresolutions,thenthepairof linearequations is said tobe consistentotherwise it is said tobeinconsistent.

    Thus, the systems in examples 2 and 4 are consistent, while pair of linearequations in example3 is inconsistent. In example4, there are infinitelymanysolutionsofthegivenpairoflinearequations.Suchasystemiscalleddependent.(Why?)

    Thuspairofequationinexample4isconsistentaswellasdependent.

    Relationbetweencoefficientandnatureofsystemofequations

    Let us compare the coefficients of same pair of linear equations.Observe thefollowingtable.

    PairofLines Comparingratios

    GraphicalRepresentation

    Algebraicinterpretation

    4x3y6=0x+2y7=0

    41

    32 = Intersecting

    LinesAuniquesolution

    2x+3y6=04x+6y24=0

    = = = = Parallellines Nosolution

    3x2y5=06x=4y+10

    = = = = = Coincidentlines

    Infinitesolutions

    Canwesaythatforthepairoflinearequationsgivenby(examineformorepairoflinearequations)

  • 53

    a1x+b1y+c1=0

    a2x2+b2y+c2=0

    (1) If ,thelinesintersectatapointandthesolutionisunique.Thepair

    ofequationsisconsistent.

    (2) If = ,thelinesareparallelandthereisnosolutionoftheequations.

    Thepairofequationsisinconsistent.

    (3) If = = , the lines are coincident and there are infinitely many

    solutions.Inthiscasethepairofequationsisdependentandconsistent.Incaseofdependentequations,oneequationcanbeobtainedfromtheotherbymultiplying or dividing the equation by a non zero real number. Forexample, inexample4,secondequationcanbeobtainedfromthefirstbymultiplyingitby2.Wecansummarizetheaboveobservationsasfollows:

    Comparingratios

    GraphicalRepresentation

    Algebraicinterpretation

    Natureofsystem

    IntersectingLines

    A uniquesolution

    Consistent

    Independent

    = Parallellines Nosolution Inconsistent

    Independent

    = = Coincidentlines

    Infinitesolutions Consistent

    Dependent

  • 54

    Example5:Forwhatvalueofpthefollowingpairofequationshasauniquesolution.2x+py=53x+3y=6Solution:Weknowthatapairoflinearequationshasauniquesolution

    If

    Herea1=2,b1=p

    a2=3,b2=3

    Since, foruniquesolution

    Sop2.

    Forallrealnumbersexcept2,thegivenpairoflinearequationswillhaveauniquesolution.

    Example6:Findthevalueofkforwhichthepairofequations2xky+3=04x+6y5=0 representparallellines.Solution:Here a1=2, b1=k, c1=3a2=4, b2=6, c2=5Thepairofequationsrepresentparallellinesif

    =

    i.e. = whichimplies

    = and

  • 55

    k=3and

    Fork=3thegivenpairoflinearequationswillrepresentparallellines.

    Example7:Forwhatvalueofk, thepairofequations3x+4y+2=0and9x+12y+k=0representcoincidentlines.SolutionHere,a1=3, b1=4, c1=2a2=9, b2=12, c2=kThepairofequationsrepresentcoincidentlinesif

    = =

    = =

    i.e. = ork=6

    Fork=6thegivenpairoflinearequationswillrepresentcoincidinglines.

    Example8:Forwhatvalueofpwillthefollowingpairoflinearequationshaveinfinitelymanysolutions?

    Px+3y(p3)=012x+pyp=0

    Solution:Here a1=p, b1=3, c1=(p3)a2=12, b2=p, c2=pApairoflinearequationshasinfinitelymanysolutionsif

  • 56

    = =

    = =

    = or =

    , 0

    P2=36or3=p3, 0P=6or p=6So,requiredvalueofpis6asp=6satisfyboththegivenconditions.Forp=6givenpairoflinearequationshaveinfinitelymanysolutions.

    3.5Algebraicmethodofsolvingpairoflinearequationsintwovariables

    Thegraphicalmethodof solvingpairof linearequationsprovidesaquickvisualization;ofsolution(s)butithasitslimitationsalso.Sometimes,itmaybe difficult to read the exact values of the coordinates of the point ofintersection fromthegraphsparticularlywhencoordinatesdonot involve

    integralvaluesbutrationalvalueslikesay( , )orirrationalvalueslike

    (2, 3)etc.

    To overcome this difficulty, we find the solution(s) of the equations byusingalgebraicmethodsasexplainedbelow:

    SubstitutionMethod

    Thismethod isusefulforsolvingapairof linearequations intwovariableswhereonevariablecaneasilybewrittenintermsoftheothervariable.3

  • 57

    Algorithmofsubstitutionmethod:

    Step1:Takeoneoftheequationsandexpressonevariablesayyintermsofx. Step2: Substitutethevalueofyobtainedinstep1intheotherequation. Step3: Simplifytheequationobtainedinstep2andfindthevalueofxbysolving

    thisequation. Step4: Substitutethevalueofxobtainedinstep3intoeitherequationandsolve

    forsecondvariable. Step5: Checktheobtainedsolutionbysubstitutingthevaluesofxandyinboth

    theoriginalequations.Example9:solvethefollowingpairoflinearequationsbysubstitution

    2x+y=5 (i)3x+2y=8 (ii)

    METHODStep1:Takeoneoftheseequationsandexpressonevariablesayyintermsofx. y=52x from(i) Step2:Substitutethevalueofyobtainedinstep1intheotherequation Substitute52xforyinequation

    (ii) 3x+2y=8 3x+2(52x)=8Step3:Simplifytheequationobtainedinstep2andfindthevalueofxbysolving

    thisequation. 3x+104x=8 x=810 x=2 x=2Step4:Substitutethevalueofxobtainedinstep3intoeitherequationandsolve

    forsecondvariable.Substitute2forxinequation(i)

  • 58

    2x+y=5 2(2)+y=5 y=54=1

    Therefore,x=2,y=1isasolutionofthegivenpairoflinearequationsintwovariables.

    Step5: Checktheobtainedsolutionbysubstitutingthevaluesofxandyinboththeoriginalequations.

    Check:For(i)equation L.H.S.=2x+y=2(2)+1=4+1=5=R.H.S.

    for(ii)equation L.H.S.=3x+2y=3(2)+2(1)=6+2=8=R.H.S. Example10:Solvethepairofequations 3x+2y=12 (i)

    4xy =5 (ii)Solution:Fromequation(ii),wehave y=4x5 Substitutingthevalueofyinequation(i),weget 3x+2(4x5)=12 3x+8x10=12 11x=22 x=2 Substitutingthevalueofxinequation(ii),weget 4 2y=5 y=85=3 So,thesolutionofthegivenequationsisx=2,y=3Check: Check:Forequation(i) L.H.S.=3x+2y=3(2)+2(3)=6+6=12=R.H.S. forequation(ii) L.H.S.=4xy=4(2)3=83=5=R.H.S

  • 59

    Example11:Usingsubstitutionmethod,solvethegivenpairofequations x+2y=4 (i)

    2x+3y =5 (ii)Solution: Thegivenequationsare x+2y=4 (i)

    2x+3y =5 (ii)fromequation(i),wehave

    x=42ySubstitutingthisvalueofxinequation(ii),weget

    2(42y)+3y=5 84y+3y=5 y=3 Substitutingthevalueofyinequation(i),weget x+2 (3)=4 x=46 x=2 Therefore,thesolutionisx=2,y=3Check: Check:Forequation(i), L.H.S.=x+2y=2+2(3)=2+6=4=R.H.S. Forequation(ii) L.H.S.=2x+3y=2(2)+3(3)=4+9=5=R.H.S. Example 12: Solve the given pair of linear equations, using the substitution

    method. x5y=7 (i)

    2x10y=5 (ii)Solution:Fromequation(i),weget x=7+5ySubstitutingthisvalueofxinequation(ii),weget 2(7+5y)10y=5

    14+10y10y=514=5(whichisnotpossible)

    Thisisafalsestatement.

  • 60

    So,thepairoflinearequationshasnosolutions.Note:observethat

    Example13:Solvethegivenpairoflinearequations 2x+3y=6 (i)

    6x+9y=18 (ii)Solution:Fromequation(i),wehave

    x=

    or,x=3

    Substitutingthisvalueofxinequation(ii),weget

    6(3 )+9y=18

    189y+9y=1818=18

    This is a true statement but contains no variables. The pair of linearequationsisdependentwithinfinitelymanysolutions.

    Observethat

    Note: Themethodof substitution isalso knownasmethodof eliminationbysubstitution.

    EliminationMethod

    Inthismethod,weeliminateoneofthetwovariablesbyequatingthecoefficienttoobtainasingleequationinonevariablewhichgivesthevalueofonevariable.Weexplainthismethodthroughanexample.

  • 61

    Algorithmofeliminationmethod:

    Step1:Writebothequationsintheformax+by=c

    Step2:Makethecoefficientsofoneofthevariablessayx,numericallyequalbymultiplyingtheequationsbysuitablerealnumber.

    Step3:Addor subtract theequationsobtained in step2 togetanequation inonlyonevariable.

    Step4:Solvetheequationforthevariableobtained.

    Step5:Substitutethevalueofthisvariable ineitherofthegivenequationsandfindthevalueoftheothervariable.Step6:Checkthesolutionfor(x,y)bysubstitutingitintheoriginalequation.Note:Whilesolvingtheequation, ifweobtainatruestatement instep3above,the system of equations has infinitelymany solutions and ifwe obtain a falsestatement,thesystemhasnosolutions.

    Youcancheck these factsbyverifying and respectively.

    Example14:Solvethepairofequations

    2x+5y+5=0 (i)

    2y=3x+17 (ii)

    Solution:

    Step1:Writebothequationsintheformax+by=c

    Rewritetheequationsas

    2x+5y=5 (i)

    3x+2y=17 (ii)

  • 62

    Step2:Makethecoefficientsofoneofthevariablessayx,numericallyequalbymultiplyingtheequationsbysuitablerealnumber.

    L.C.M. of coefficients (2 and 3) of x is 6.Wemultiply equation (i) by 3 andequation(ii)by2

    6x+15y=15 (iii)

    6x+4y=34 (iv)

    Step3:Addor subtract theequationsobtained in step2 togetanequation inonlyonevariable.

    Addingequations(iii)and(iv),weget

    19y=19

    Step4:Solvetheequationforthevariableobtained.

    y= 1

    Step5:Substitutethevalueofthisvariable ineitherofthegivenequationsandfindthevalueoftheothervariable.

    Substitute1foryinequation(i) 2x+5y=5 2x+5(1)=5 2x=10 x=5 Thesolutionisx=5,y=1

    Step6:Checkthesolutionfor(x,y)bysubstitutingitintheoriginalequation.

    Note:Whilesolvingtheequation, ifweobtainatruestatement instep3above,the system of equations has infinitelymany solutions and ifwe obtain a falsestatement,thesystemhasnosolutions.

  • 63

    Youcancheck these factsbyverifying and respectively.

    Example15:Solvethepairofequations x+y=3 (i)

    2x+5y=12 (ii)Solution: Toeliminatex,wemultiplyequation(i)by2andget

    2x+2y=6 (iii) Subtractingequation(iii)fromequation(ii),weget 2x+5y=12 2x+2y=6

    3y=6 y=2 Puttingthevalueofyinequation(i),weget x+2=3 x=32=1 So,solutionoftheequationsisx=1,y=2

    Alternatively,toeliminatey,wemultiplyequation(i)by5andget

    5x+5y=15 (iv)Subtractingequation(iii)fromequation(ii),weget

    5x+5y=15 2x+5y=12

    3x=3 x=1 Puttingthevalueofxinequation(i),weget

    1 +y=3y=2

    So,solutionoftheequationisx=1,y=2.

  • 64

    Example16:Solvethepairofequations 31x+43y=117 (i)

    43x+31y=105 (ii)Solution: Wearegiven

    31x+43y=117 (i)43x+31y=105 (ii)

    Ifwemultiplyequation(i)by43andequation(ii)by31,thencalculationbecomestediousandtimeconsumingbecauseoflargenumbersintheproducts.

    Toavoidthis,weadoptthefollowingprocedure:

    Addingequation(i)and(ii),weget

    74x+74y=222 Orx+y=3 (iii)

    Subtractingequation(i)fromequation(ii),weget 12x12y=12 xy=1 (iv)

    Again,addingequation(iii)and(iv),weget 2x=2 x=1

    Substitutingx=1in(iii),weget 1+y=3 Ory=2

    So,solutionoftheequationsisx=1,y=2Check:

    Forequation(i)L.H.S.=31x+43y=31(1)+43(2)=31+86=117=R.H.S.

    Forequation(ii)

    L.H.S.=43x+31y=43(1)+31(2)=43+62=105=R.H.S. Note: Theprocedure ispossiblewhencoefficientofx inoneequation isthesameascoefficientofyinotherequationandviceversa.

  • 65

    CrossMultiplicationmethodAlgorithmofcrossmultiplicationmethod: Tosolvethepairofequations x+ + =0 x+ + =0 Step1:Writethecoefficientsofxandyandtheconstanttermsinthefollowingmanner. x y = 1 b1 c1 c1 a1 a1 b1 b2 c2 c2 a2 a2 b2 i.ewritecoefficientofyandconstanttermsbelowx.writeconstanttermsandcoefficientsofxbelowy.writecoefficientsofxandybelow1. x y = 1 5 2 2 1 4 5 4 1 1 3 3 4Step2:Multiplynumberswrittenateacharrow,positivesignwitharrowpointingdownwardandnegativesignwitharrowpointingupward.

    Step3:Taking

    And

    Clearlyif,a1b2a2b10,thenonlywecanfindxandyi.e.Uniquesolution.

  • 66

    Note:Ifa1b2a2b1=0,thenthismethodisnotapplicable.Thenthesystemmaybe having either no solution or infinitelymany solutions. One can find actualstatusoftheequationsbycomparingtheconstants.

    Example17:Solve the followingpairof linearequationsbycrossmultiplicationmethod:

    2x+5y=17

    5x+3y=14

    Solution:Wecanrewritethegivenequationsas 2x+5y17=0 (i) 5x+3y14=0 (ii) Herea1=2,b1=5 a2=5b2=3 a1b2a2b1=2(3)5(5)=625=190So,thesystemhastheuniquesolution.Bycrossmultiplicationmethod: x y = 1 5 17 17 2 2 5

    3 14 14 5 5 3

    Or

    Orx=1andy= 3

    Check:Youareadvisedtocheckthesolutionforx=1,y=3.

  • 67

    Example18:Solvethefollowingpairoflinearequations 3x+2y=13 6x+4y=10 Solution:Thegivenpairoflinearequationscanbewrittenas 3x+2y13=0 (i) 6x+4y10 =0 (ii) Herea1=3,b1=2,a2=6,b2=4 a1b2a2b1=3 46 2=1212=0

    Since,a1b2a2b1=0,wecannotapplycrossmultiplicationmethodinsolvingtheseequations.Checkthat

    12

    36

    12

    , 12

    24

    12

    , 12

    1310

    .

    , .

    Applicationsoflinearequationsinreallifesituations

    Wecansolvevariousdaily lifeproblemsbyconvertingthem intoapairof linear

    equationsintwovariablesandthensolvingthepairofequations.

    Example19:

    Sumoftwonumbersis42andtheirdifferenceis12.Findthenumbers.

    Solution:

    Letthenumbersbexandy

    Sumofthenumbersis42(given)

    So,x+y=42(i)

    Differenceofthenumbersis12

    So,xy=12(ii)

    oryx=12(iii)

  • 68

    Adding(i)and(ii),weget

    2x=54

    x=54/2=27

    Substitutinginequation(i)

    27+y=42

    y=4227=15

    Solving(i)and(iii)weget

    x=15,y=27

    Thus,inbothcasenumbersare15,27.

    Check: Sumofnumbers.15+27=42

    Differenceofnumbers=2715=12.

    Example20:

    Thesumofdigitsofatwodigitnumberis9.Alsoninetimesthisnumberistwice

    thenumberobtainedbyreversingtheorderofthenumber.Findthenumber.

    Solution:

    Lettheunitsdigitbexandthetensdigitbey.

    Therequirednumber=10y+x

    Onreversingtheorderofdigits

    Thenumber=10x+y

    Thesumofdigitsofatwodigitnumberis9

    So,x+y=9(i)

    Ninetimesthenumberistwicethenumberobtainedbyreversingtheorderof

    thenumber,so,

    9(10y+x)=2(10x+y)

    Or 90y+9x=20x+2y

  • 69

    Or 11x+88y=0

    Or x8y=0(ii)

    Subtracting(ii)from(i)

    x+y=9

    x8y=0

    +

    9y=9

    y=1

    Substitutingthevalueofyinequation(i)

    x+1=9

    x=91=8

    Thus,therequirednumber=10y+x

    =10(1)+8=18

    Check

    1+8=9,sumofdigitis9

    9(18)=162=2(81)whichistrue

    Example21:

    Thepresentageoffatherisfourtimestheageofhisson.Fouryearslater,hewill

    bethreetimestheageofhisson.Findtheirpresentages.

    Solution:

    Letthepresentsageoffatherbexyears

    Andthatofsonbeyyears

    Presentageoffatherisfourtimestheageofhisson

    So,x=4y(i)

    Fouryearslater,fathersagewillbe(x+4)yearsandsonsagewillbe(y+4)years

  • 70

    Accordingthequestion

    (x+4)=3(y+4)

    Orx+4=3y+12

    x=3y+8(ii)

    Fromequation(i)andequation(ii)

    4y=3y+8

    Ory=8

    Substitutingthevalueofyinequation(i),weget

    x=4 8

    x=32

    Thus,thefathersageis32yearsandthesonsageis8years

    Check: 32=4(8)

    Also 32+4=36

    8+4=12

    So, 3(12)=36

    Example22:

    Thepresentageofafatherexceedsthesumoftheagesofhisthreechildrenby8

    years. Aftertenyearshisagewillbecome ofthesumofagesofthechildren.

    Findthepresentageofthefather.

    Solution

    Letthepresentageoffather=xyearsand

    letthesumofagesofthreechildren=yyears.

    After10years,fathersage=(x+10)years.

    TheSumofagesof3children=(y+3X10)(why?)

    =(y+30)years.

  • 71

    Accordingtotheproblem,

    x=y+8 (i)

    Also, x+10= (y+30)

    6x+60=5y+150(ii)

    Substitutingvalueofxfromequation(i)intoequation(ii)

    6(y+8)+60=5y+150

    6y+48+60=5y+150

    6y5y=150108

    y=42

    Substitutingthevalueofyinequation(i)

    x=42+8=50

    So,thepresentageoffather=50years

    (Youcancheckthesolution)

    Example23:

    InaABC, C=3 Aand B=2( A+ C)theanglesofthetriangle.

    Solution

    Let A=x, C=y

    C=3 A

    y=3x(1)

    Also B=2( A+ C)

    =2(x+y)

    Byanglesumproperty

    A+ B+ C=180

    x+2(x+y)+y=180

    3x+3y=180

  • 72

    or x+y=60(2)

    Substitutingvalueofyfromequation(1)inequation(2)

    x+(3x)=60

    x+3x=60

    4x=60

    x==15

    Substitutionvalueofxinequation(1)

    y=3X15=45

    So, A=15, C=45

    B=2( A+ C)=2(15+45)=2X60=120

    Example24:

    Theareaofarectanglegets reducedby80squnits if its length isreducedby5

    unitsandthebreadthisincreasedby2units.Ifweincreasethelengthby10units

    anddecreasethebreadthby5unitstheareaisincreasedby50sq.units.Findthe

    lengthandbreadthoftherectangle.

    Solution:

    Letthelengthofrectangle= xunits

    breadthofrectangle= yunits

    Areaoftherectangle= xysq.units

    Accordingto1stcondition

    (x5)(y+2)= xy80

    xy=2x+5y10=xy80

    2x5y=70(i)

    Accordingtosecondcondition,

    (x+10)(y5)=xy+50

  • 73

    xy5x+10y50=xy+50

    5x+10y=100

    orx2y=20(ii)

    orx=2y20

    Substituting2y20forxinequation(i)

    2(2y20)5y=70

    4y405y=70

    y=30

    y=30

    Substitutionthevalueofyinequation(ii)

    x2(30)=20

    x=20+60=40

    Thus,thelengthoftherectangle =40units

    Thebreadthoftherectangle =30units

    WORKRATIOPROBLEM

    Example25

    8menand12womencan finishapieceofwork in10dayswhile6menand8

    womencan finish it in14days. Findthetimetakenbyonemanaloneto finish

    thework.Also,findthetimetakenbyonewomentofinishthework.

    Solution:

    Letthetimetakenbyonemanalonetofinishthework=xdays

    Onedayworkofoneman=

    Letthetimetakenbyonewomanalonetofinishthework=ydays

    Onedayworkofonewoman=

  • 74

    1dayworkof8man=

    1dayworkof12woman=

    8men&12womancanfinishtheworkin10days

    10 1

    1(i)

    6men&12womencanfinishtheworkin14days

    14 1

    1(ii)

    Substituting ,

    80u+120v=1

    84u+112v=1

    Solvingusingcrossmultiplicationmethod

    120 112 84 80

    1

    80 112 82 120

    8

    1120

    1140

    4

    1120

    1280

    140 280

    Onemanalonecanfinishtheworkin140days

    Onewomanalonecanfinishtheworkin280days

    Youcanuseanymethodtosolvetheseequations.

  • 75

    Speedtime

    Example26:

    PointAandBare100kmapartonahighway.OnecarstartsfromAandanother

    from B at the same time. If the cars travel at a constant speeds in the same

    directiontheymeetinahour.Findthespeedofthetwocars.

    Solution:

    LetthespeedofcarstartingfrompointA=xKm/h

    LetthespeedofcarstartingfrompointB=yKm/h

    DistancebetweenAandB=100km

    LetthemeetatP

    Whentheytravelintheoppositedirections,

    DistancetravelledbycarstationfromA=1

    DistancetravelledbycarstationfromB=1

    x+y=100(i)

    Whentheytravelinthesamedirection.LettheymeetatQ

    Distancetravelby1stcarAin5hour=5x=AQ

    Distancetravelby2ndcarB=5y=BQ

    sinceAQBQ=100

    so, 5x+5y=100

    or x+y=20(ii)

  • 76

    solvingequation(i)and(ii)byadding,weget

    2x=120

    x=60

    Puttingx=60in(i),weget

    60+y=100

    y=40

    So,thespeedofcarstartingfrompointA=60Km/h

    ThespeedofcarstartingfrompointB=40Km/h

    Example27:

    Apersoncanrowdown20kmin2hoursandupstream4kmin2hours.Findhis

    speedofrowinginstillwaterandspeedofthestream.

    Solution:

    Letthespeedofrowinginstillwater=xkm/h

    Andspeedofstream=ykm/h

    Speeddownstream=(x+y)mk/h

    Speedupstream=(xy)km/h

    Hecanrowdown20kmin2hours

    202

    2(x+y)=20

    x+y=10(i)

    Hecanrowupstream4kmin2hours

    So, 2

    2(xy)=4

    xy=2(ii)

  • 77

    addingequation(i)and(ii)

    2x=12

    x=6

    Substitutingthevalueofxinequation(i),weget

    6+y=10

    y=4

    Thus, the speedof rowing in stillwater=6km/hand speedof the stream=4

    km/h(Checkthesolutionwiththeoriginalquestion)

  • 78

    SSTTUUDDEENNTTSS

    SSUUPPPPOORRTT

    MMAATTEERRIIAALL

  • 79

    StudentsWorksheet1

    Warmup(W1)

    Nameofthestudent:___________________ Date:__________

    Inwhichquadrantthegivenorderedpairswilllie?

    (0,3)

    (2,3)

    (0,2) (2,3)

    (4,0)

    (5,7) (5,7)

    (2,3)

    (0,0)

    OrderedPair Location

  • 80

    SelfAssessmentRubric1WarmUp(W1)

    Parameter

    Locationofapointinfourquadrants

    Locationofapointonthexaxis

    Locationofapointontheyaxis

    LocationoftheOrigin.

  • 81

    StudentsWorksheet2

    Warmup(W2)

    Nameofthestudent:___________________ Date:__________

    Tellwhichofthegivenlinearequationsareinonevariableandwhichareintwovariables?Namethevariablesalso.Readthecoefficientsofthevariables.

    (i) 2xy=5(ii) yz=6

    (iii) 54x=7z(iv) 3 +8=9x(v) 7 3 0(vi) 5x+3y=4x2y(vii) 4x3y(viii) 3x=y(ix) 3x=0(x) 4y(xi) 5y3x=0

  • 82

    SelfAssessmentRubric2WarmUp(W2)

    Parameter

    Can recognize linearequation

    inonevariable.

    Can identifya linearequation

    intwovariables

  • Nameo

    1.Plott

    ABCDEF

    ofstudent_

    thegiveno

    A. (1,3)B. (2,5)C. (3,2)D. (4,1). (3,0). (0,2)

    _________

    orderedpa

    Student

    Warm

    _________

    airsonthe

    83

    sWorkshe

    mup(W3

    ___

    sameCart

    eet3

    3)

    D

    tesianplan

    Date:____

    ne.

    _______

  • 84

    2.Determinethecoordinatesofeachofthepointsshowninthefigure

    Points CoordinatesA

    B

    C

    D

    E

    F

    G

    P

    Q

  • 85

    SelfAssessmentRubric3Warmup(W3)

    Parameter

    Abletoplot/locateanordered

    pairinallfourquadrants

    Abletoplot/locateapointon

    thexaxis

    Abletoplot/locateapointon

    theyaxis

  • 86

    StudentsWorksheet4

    PreContent(P1)

    Nameofthestudent:___________________ Date:__________

    Readthegiveninstructionsandwriteyouranswer

    Writealinearequationintwovariables

    1. Withcoefficientofxlessthan3andcoefficientofymorethan5

    2. Withcoefficientofxless3andcoefficientofylessthan5

    3. Withcoefficientofxequalto8andcoefficientofymorethan8

    4. Withcoefficientofxandcoefficientofymorethan5

    5. Withsumofthecoefficientsofaandyas2andconstantterm3

  • 87

    SelfAssessmentRubric4PreContent(P1)

    Parameter

    Has knowledge of terms

    coefficients, constant term,

    variable

    Abletowritealinearequation

    in two variable with given

    coefficients

  • 88

    StudentsWorksheet5

    PreContent(P2)

    Nameofthestudent:___________________ Date:__________

    1. Fillintheabscissaortheordinatesothatthegivenorderedpairisasolutionoftheindicatedequation.

    (1)x+3y=5 (,2)(2)2x3y=7 (4,)

    (3)3x+2y=7 ( ,)

    (4)x2y=6 (0,)

    2.Completetheorderedpairsothatallofthemaresolutionsof

    2x+3y=6

    (1)(3,)

    (2)(0,)

    (3)(4,)

    (4)(,1)3.Thegraphoftheequationx+y=10isa..

    (a) Curvedline(b) Straightline(c) StraightlinePassingthroughorigin(d) Straightlineparalleltoxaxis

  • 89

    SelfAssessmentRubric5PreContent(P2)

    Parameter

    Abletofindsolutionofagiven

    linear equation in two

    variables

    Able to verify that a given

    point is a solution of linear

    equationintwovariables

    Know that the graph of a

    linear equation in two

    variablesisastraightline

  • 90

    StudentsWorksheet6,PlottingonCartesianPlane

    Content(CW1)

    Nameofthestudent:___________________ Date:__________

    1. Plotthecoordinatesgiveninthefollowingtableandjointhem.

    X O 2 2Y 5 0 10

    Whatdoweget? ________________________________Isitalineorlinesegment? ________________________________

  • 91

    Locatetwomorepointsonit. ________________________________Howmanymorepointscanyoulocateontheline? ________________________________

    2. PlotthecoordinatesgiveninthefollowingtablesonthesameCartesianplane.

    X 1 O 3Y 2 3 0

    Whattypeoflinesarethey? ________________________________Dothelineshaveanypointcommon? ________________________________

    x 2 4 6y 0 3 6

  • 92

    Ifyes,howmany? ________________________________

    3. Representgraphicallythepairoflinearequationsintwovariables:2x+y=44x+2y=8Completethetablesanddrawthegraph.

    X 1 1 Y 2

    Whattypesoflineareobtained? _______________________________________Howmanypointsarecommoninthetwolines? _______________________________________

    X 0 2

    Y 0

  • 93

    4. Representgraphicallythepairoflinearequationsintwovariables:

    x2y=22x+4y=4

    X Y

    X Y

  • 94

    SelfAssessmentRubric6Content(CW1)

    Parameter

    Abletoplotthegivenordered

    pairs

    Abletoobservethegraphasa

    straightline

    Able to plot a pair of linear

    equations in two variablesby

    makingtableofvalues

    Abletorepresentintersecting,

    parallel or coincident lines

    usingpairofequations

  • 95

    StudentsWorksheet7,SolutionofPairofLinearEquationsinTwoVariablesGraphically

    ContentWorksheet(CW2)

    Nameofthestudent:___________________ Date:__________

    1. Givenbelowisthegraphofapairoflinearequationintwovariables.Observeitandanswerthequestionsgiven.

    Whattypesoflinesareobtained? ________________________________

    Howmanypointsarecommoninthetwolines? ________________________________

    Does(3,1)lieonboththeequations?Explain

    ___________________________________________________________________

  • 96

    2. Representgraphicallythegivenpairoflinearequationsintwovariables:

    xy=2,x+y=6

    Maketableofvaluesforboththeequations

    xy=2x+y=6

    X 0 1Y 0

    Weobservethelinesintersectat

    Thepairofequationshasaunique/infinite/nosolution(s)

    Verifythatthepointofintersectionliesonbothlines.

    X 0 1Y 0

  • 97

    3.Drawthegraphsofthefollowingpairofequations

    2x+y=2

    2x+y=8

    Tableofvaluesfor2x+y=2

    x y

    Tableofvaluesfor2x+y=8

    X Y

    Thelinesobtainedare _______________________________________

    Thelineshave____________________________pointsincommon.

  • 98

    Thegivenpairofequationhas___________________________________solution.

    5. Solvegraphicallythelinearequationsbytakingsuitablevaluesofxandy.2x+5y=1 4x+10y=2

    X Y

    Whatdoyouobserve? _____________________________________________________

    Thegivenpairoflinearequationshas____________solution(s).

    X Y

  • 99

    Brainstorming:

    Whattypesofgraphsarepossible?

    Whattypesofsolutionsarepossible?

    SelfAssessmentRubric7Content(CW2)

    Parameter

    Able to plot pair of linear

    equationsongraph

    Able to observe the types of

    graphs

  • 100

    StudentsWorksheet8,

    SolutionofPairofLinearEquationsinTwoVariablesGraphically

    ContentWorksheet(CW3)

    Nameofthestudent:___________________ Date:__________

    1. Givenalinearequationin2variables,identifythefollowing.

    S.No. Linearequation Coefficientofx Coefficientofy Constantterm

    1.

    2.

    3.

    2x+3y7=0

    3x2y7=0

    4x+5y+14=0

    2. Task1

    Investigatethegivenpairsofequations.Findtheratiosofcoefficientsofx,coefficientsofyandconstantterms.Writeyourobservations.Drawtheirrespectivegraphsusing(GeoGebra)andexpressyourresultintermsofconditionsforconsistencyorinconsistency.

    a1x+b1y=c1

    a2x+b2y=c2

    Relationbetweentheobservedratios

    1. 11 6 3,

    22 12 17

    2. 9 5 0,

    27 15 4

  • 10