Writing-and-Graphing-Linear-Equations

WritingandGraphingLinearEquationsLinearequationscanbeusedtorepresentrelationships.WritingEquationsandGraphing•Theseactivitiesintroduceratesofchangeanddefinesslopeofalineastheratiooftheverticalchangetothehorizontalchange.•Thisleadstographingalinearequationandwritingtheequationofalineinthreedifferentforms.Linearequation–Anequationwhosesolutionsformastraightlineonacoordinateplane.Collinear–Pointsthatlieonthesameline.Slope–Ameasureofthesteepnessofalineonagraph;risedividedbytherun.Alinearequationisanequationwhosesolutionsfallonalineonthecoordinateplane.Allsolutionsofaparticularlinearequationfallontheline,andallthepointsonthelinearesolutionsoftheequation.Lookatthegraphtotheleft,points(1,3)and(-3,-5)arefoundonthelineandaresolutionstotheequation.Ifanequationislinear,aconstantchangeinthex-valueproducesaconstantchangeinthey-value.Thegraphtotherightshowsanexamplewhereeachtimethex-valueincreasesby2,they-valueincreasesby3.Theequationy=2x+6isalinearequationbecauseitisthegraphofastraightlineandeachtimexincreasesby1unit,yincreasesby2XY=2x+6Y(x,y)12(1)+68(1,8)22(2)+610(2,10)32(3)+612(3,12)42(4)+614(4,14)52(5)+616(5,16)RealworldexampleThegraph(c=5x+10)attheleftshowsthecostforCompanyAcellphonecharges.WhatdoesCompanyAchargefor20minutesofservice?Graphingequationscanbedownseveraldifferentways.Tablescanbeusedtographlinearequationsbysimplygraphingthepointsfromthetable.Completethetablebelow,thengraphandtellwhetheritislinear.xy=2x+3y(x,y)-2-1012Canyoudetermineiftheequationislinear?Theequationy=2x+3isalinearequationbecauseitisthegraphofastraightline.Eachtimexincreasesby1unit,yincreasesby2.Xy=2x+3Y(x,y)-22(-2)+3-1(-2,1)-12(-1)+31(-1,1)02(0)+33(0,3)12(1)+35(1,5)22(2)+37(2,7)SlopeRateofchangeSlopeofalineisitsrateofchange.Thefollowingexampledescribeshowslope(rateofchange)isapplied.Rateofchangeisalsoknowasgradeorpitch,orriseoverrun.ChangeisoftensymbolizedinmathematicsbyadeltaforwhichthesymbolistheGreekletter:ΔFindingslope(rateofchange)usingagraphandtwopoints.Ifanequationislinear,aconstantchangeinthex-valuecorrespondstoaconstantchangeinthey-value.Thegraphshowsanexamplewhereeachtimethex-valueincreasesby3,they-valueincreasesby2.Slopes:positive,negative,noslope(zero),undefined.Remember,linearequationshaveconstantslope.Foralineonthecoordinateplane,slopeisthefollowingratio.Thisratioisoftenreferredtoas“riseoverrun”.Findtheslopeofthelinethatpassesthrougheachpairofpoints.1)(1,3)and(2,4)2)(0,0)and(6,-3)3)(2,-5)and(1,-2)4)(3,1)and(0,3)5)(-2,-8)and(1,4)GraphingaLineUsingaPointandtheSlopeGraphthelinepassingthrough(1,3)withslope2.Giventhepoint(4,2),findtheslopeofthisline?Tomakefindingslopeeasier,findwherethelinecrossesatanxandyjunction.FindingSlopefromaGraphUsethegraphofthelinetodetermineitsslope.Choosetwopointsontheline(-4,4)and(8,-2).Counttheriseoverrunoryoucanusetheslopeformula.Noticeifyouswitch(x1,y1)and(x2,y2),yougetthesameslope:Usethegraphtofindtheslopeoftheline.UsingSlopesandInterceptsx-interceptsandy-interceptsx-intercept–thex-coordinateofthepointwherethegraphofalinecrossesthex-axis(wherey=0).y-intercept–they-coordinateofthepointwherethegraphofalinecrossesthey-axis(wherex=0).Slope-interceptform(ofanequation)–alinearequationwrittenintheformy=mx+b,wheremrepresentsslopeandbrepresentsthey-intercept.Standardform(ofanequation)–anequationwrittenintheformofAx+By=C,whereA,B,andCarerealnumbers,andAandBareboth≠0.StandardFormofanEquation•Thestandardformofalinearequation,youcanusethex-andy-interceptstomakeagraph.•Thex-interceptisthex-valueofthepointwherethelinecrosses.•They-interceptisthey-valueofthepointwherethelinecrosses.Ax+By=CTographalinearequationinstandardform,youfinethex-interceptbysubstituting0foryandsolvingforx.Thensubstitute0forxandsolvefory.2x+3y=62x+3(0)=62x=6x=3Thex-interceptis3.(y=0)2x+3y=62(0)+3y=63y=6y=2They-interceptis2.(x=0)Let’stakealookatthatequationagain!2x+3y=62x+3y=62x=63y=6x=3y=2Sinceyouaresubstituting(0)inforonevariableandsolvingfortheother,anynumbermultipliedtimes(0)=0.So,inthefirstexample3(0)=0,andinthesecondexample2(0)=0.Since3(0)=0,justcoverupthe3yandsolvewhat’sleft.Again,since2(0)=0,justcoverup2xandsolvewhat’sleft.Findthex-interceptandy-interceptofeachline.Usetheinterceptstographtheequation.1)x–y=52)2x+3y=123)4x=12+3y4)2x+y=75)2y=20–4xSlope-interceptFormy=mx+bSlope-interceptForm•Anequationwhosegraphisastraightlineisalinearequation.Sinceafunctionruleisanequation,afunctioncanalsobelinear.•m=slope•b=y-interceptY=mx+b(ifyouknowtheslopeandwherethelinecrossesthey-axis,usethisform)Forexampleintheequation;y=3x+6m=3,sotheslopeis3b=+6,sothey-interceptis+6Let’slookatanother:y=4/5x-7m=4/5,sotheslopeis4/5b=-7,sothey-interceptis-7Pleasenotethatintheslope-interceptformula;y=mx+bthe“y”termisallbyitselfontheleftsideoftheequation.Thatisveryimportant!WHY?Ifthe“y”isnotallbyitself,thenwemustfirstusetherulesofalgebratoisolatethe“y”term.Forexampleintheequation:2y=8x+10Youwillnoticethatinordertoget“y”allbyitselfwehavetodividebothsidesby2.Afteryouhavedonethat,theequationbecomes:Y=4x+5Onlythencanwedeterminetheslope(4),andthey-intercept(+5)OK…gettingbac