Name:_________________________ Math________,Period__________Mr.Rogove Date:__________

G8M4:StudyGuideSystemsofLinearEquations1

Study Guide: Systems of Linear Equations

Systems of Linear Equations

Asystemoflinearequationsiswhentwoormorelinearequationsareinvolvedinthesameproblem.Thesolutionforasystemoflinearequationsistheorderedpair(!,!)thatmakesBOTHequationstrue.Solutions to systems of linear equations:Systemsofequationseitherhaveoneuniquesolution,nosolution,orinfinitelymanysolutions.OneUniqueSolution:Iftwolinearequationshavedifferentslopes,therewillbeoneuniquesolution,whichisatthepointofintersection.NoSolutions:Iftwolinearequationshavethesameslope,butdifferenty-intercepts,thelinesareparallelandthereisNOsolutiontothesystemofequationbecausethereisnopointofintersection.InfinitelyManySolutions:IftwolinearequationshavethesameslopeANDthesamey-intercept,thetwolinesareidenticalandthesystemwillhaveinfinitelymanysolutions.

Solving Systems of Equations: Graphing Onacoordinateplane,theintersection(coordinatepair)oftwolinesisthesolutiontoasystemoflinearequations.Example:Thesolutiontothesystemofequationsbelowis(1, 2)

Theequationfor!! is ! = 2!.Substitute(1,2)intothelinearequationtomakesurethatthepointISontheline.Theequationfor!! is ! = −! + 3.Substitute(1,2)intothelinearequationtomakesurethatthepointISontheline.

!!!!

Name:_________________________ Math________,Period__________Mr.Rogove Date:__________

G8M4:StudyGuideSystemsofLinearEquations2

Solving Systems of Equations: Substitution

Systemsofequationscanbesolvedbysubstitutinganexpressionfromoneequationintotheotherequation.Examples:1.

! = 32 ! − 4

! = −! + 9

Sinceyisequaltoboth(!! ! − 4) !"#(−! + 9),thosetwoexpressionsareequaltoeachother…

32 ! − 4 = −! + 9

Tosolvethesystem,solveforx,andthensubstituteyoursolutionforxintooneofyouroriginalequationsandsolvefory.

2.! = 2! − 13! + 2! = 6

Sinceyisisolatedinthefirstequation,wecansubstitute“2! − 1”whereverweseeyinthesecondequation:

3! + 2 2! − 1 = 6Tosolvethissystem,wecansolveforx,andthensubstituteyoursolutionforxintooneoftheoriginalequationsandsolvefory.

WHENTOSOLVEUSINGSUBSTITUTION:

1.Whenbothequationsarealreadyinslopeinterceptform(asinexample1above)2.Whenonevariableisisolatedinoneofthelinearequations(asinexample2above)

Solving Systems of Equation: Elimination SolvesystemsoflinearequationsusingeliminationbyADDINGequationstoeliminateavariable.NOTE:YoumightneedtomultiplyanENTIREequationbeforeyoucanaddthetwoequationstoeliminateavariable.Examples:

6! − 5! = 212! + 5! = −5

Since5! and − 5!areopposites,ifweaddthetwoequationstogether,wecaneliminatethey-variable.

8! + 0! = 16Fromhere,solveforx,andthensubstituteyoursolutionforxintooneofyouroriginalequationsandsolvefory.

6! − 4! = 182! + 3! = 4

Sincetherearenoopposites,multiplythesecondequationby−3(tocreateasituationwhereyouhaveopposites(6! and − 6!)andthenfollowthestepsintheexampleontheleft.

6! − 4! = 18−6! − 9! = −12

Name:_________________________ Math________,Period__________Mr.Rogove Date:__________

G8M4:StudyGuideSystemsofLinearEquations3

The Problem Set CompletetheproblemsbelowandturnthissheetinpriortocompletingtheassessmentDeterminewhethereachofthefollowinghaveoneuniquesolution,infinitelymanysolutionsornosolutions.DoNOTsolvethesystem,butexplainHOWYOUKNOWthenumberofsolutionsinonesentence.

! = 34 ! + 12

3! − 4! = 48

4! − 2! = −7! = 1

2 ! − 14

! = 3! − 42! + 6! = 12

7! = 14! − 214! − 2! = 6

SolvethefollowingusingEITHERsubstitutionorelimination—yourchoice.Whicheveryouchoose,writeasentenceexplainingWHYyouselectedthemethodyoudid.

! = 2! − 1! + 3! = 9

2! − 5! = 94! + 2! = 3

Name:_________________________ Math________,Period__________Mr.Rogove Date:__________

G8M4:StudyGuideSystemsofLinearEquations4

! = ! − 4! + ! = 4

! + 3! = 144! − 6! = 2

Solvethefollowingwordproblems.Bananashave105calorieseach.Appleshave75calorieseach.Inoneday,Maryonlyeatsthosetwofruits,andsheeats13piecesoffruit,andconsumesatotalof1215calories.HowmanyofeachfruitdidMaryconsume?Howdoyouthinkherstomachfeels?

OniTunes,youcandownloadsongsfor$1.29,andrentHDmoviesfor$5.99each.IfDominicspends$63.40for20totalpiecesofmedia(thisincludesbothmoviesandsongs),howmanymoviesdidherent?Howmanysongsdidhedownload?

Name:_________________________ Math________,Period__________Mr.Rogove Date:__________

G8M4:StudyGuideSystemsofLinearEquations5

Solvethefollowingsystemsofequationsbygraphing,usingsubstitution,andusingelimination.

! = − 32 ! + 13! − 4! = 12

Solvebysubstitution:Solvebyelimination:

! − ! = −1! = 3! + 2

Solvebysubstitution:Solvebyelimination: