System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.
Study Guide: Systems of Linear Equationsmrrogove.weebly.com/uploads/4/3/0/0/43009773/g8m4_sg...A...
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Name:_________________________ Math ________, Period __________ Mr. Rogove Date: __________ G8M4: Study Guide Systems of Linear Equations 1 Study Guide: Systems of Linear Equations Systems of Linear Equations A system of linear equations is when two or more linear equations are involved in the same problem. The solution for a system of linear equations is the ordered pair (, ) that makes BOTH equations true. Solutions to systems of linear equations: Systems of equations either have one unique solution, no solution, or infinitely many solutions. One Unique Solution: If two linear equations have different slopes, there will be one unique solution, which is at the point of intersection. No Solutions: If two linear equations have the same slope, but different y-intercepts, the lines are parallel and there is NO solution to the system of equation because there is no point of intersection. Infinitely Many Solutions: If two linear equations have the same slope AND the same y-intercept, the two lines are identical and the system will have infinitely many solutions. Solving Systems of Equations: Graphing On a coordinate plane, the intersection (coordinate pair) of two lines is the solution to a system of linear equations. Example: The solution to the system of equations below is (1, 2) The equation for ! is = 2. Substitute (1,2) into the linear equation to make sure that the point IS on the line. The equation for ! is = − + 3. Substitute (1,2) into the linear equation to make sure that the point IS on the line. ! !
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Study Guide: Systems of Linear Equations
Systems of Linear Equations
Asystemoflinearequationsiswhentwoormorelinearequationsareinvolvedinthesameproblem.Thesolutionforasystemoflinearequationsistheorderedpair(𝑥,𝑦)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)
Theequationfor𝑙! is 𝑦 = 2𝑥.Substitute(1,2)intothelinearequationtomakesurethatthepointISontheline.Theequationfor𝑙! is 𝑦 = −𝑥 + 3.Substitute(1,2)intothelinearequationtomakesurethatthepointISontheline.
𝑙!𝑙!
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Solving Systems of Equations: Substitution
Systemsofequationscanbesolvedbysubstitutinganexpressionfromoneequationintotheotherequation.Examples:1.
𝑦 =32 𝑥 − 4
𝑦 = −𝑥 + 9
Sinceyisequaltoboth(!!𝑥 − 4) 𝐀𝐍𝐃(−𝑥 + 9),thosetwo
expressionsareequaltoeachother…
32 𝑥 − 4 = −𝑥 + 9
Tosolvethesystem,solveforx,andthensubstituteyoursolutionforxintooneofyouroriginalequationsandsolvefory.
2.𝑦 = 2𝑥 − 13𝑥 + 2𝑦 = 6
Sinceyisisolatedinthefirstequation,wecansubstitute“2𝑥 − 1”whereverweseeyinthesecondequation:
3𝑥 + 2 2𝑥 − 1 = 6Tosolvethissystem,wecansolveforx,andthensubstituteyoursolutionforxintooneoftheoriginalequationsandsolvefory.
WHENTOSOLVEUSINGSUBSTITUTION:
1.Whenbothequationsarealreadyinslopeinterceptform(asinexample1above)2.Whenonevariableisisolatedinoneofthelinearequations(asinexample2above)
Solving Systems of Equation: Elimination SolvesystemsoflinearequationsusingeliminationbyADDINGequationstoeliminateavariable.NOTE:YoumightneedtomultiplyanENTIREequationbeforeyoucanaddthetwoequationstoeliminateavariable.Examples:
6𝑥 − 5𝑦 = 212𝑥 + 5𝑦 = −5
Since5𝑦 and − 5𝑦areopposites,ifweaddthetwoequationstogether,wecaneliminatethey-variable.
8𝑥 + 0𝑦 = 16Fromhere,solveforx,andthensubstituteyoursolutionforxintooneofyouroriginalequationsandsolvefory.
6𝑥 − 4𝑦 = 182𝑥 + 3𝑦 = 4
Sincetherearenoopposites,multiplythesecondequationby−3(tocreateasituationwhereyouhaveopposites(6𝑥 and − 6𝑥)andthenfollowthestepsintheexampleontheleft.
6𝑥 − 4𝑦 = 18−6𝑥 − 9𝑦 = −12
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The Problem Set CompletetheproblemsbelowandturnthissheetinpriortocompletingtheassessmentDeterminewhethereachofthefollowinghaveoneuniquesolution,infinitelymanysolutionsornosolutions.DoNOTsolvethesystem,butexplainHOWYOUKNOWthenumberofsolutionsinonesentence.
𝑦 =34 𝑥 + 12
3𝑥 − 4𝑦 = 48
4𝑥 − 2𝑦 = −7
𝑦 =12 𝑥 − 14
𝑥 = 3𝑦 − 42𝑥 + 6𝑦 = 12
7𝑦 = 14𝑥 − 214𝑥 − 2𝑦 = 6
SolvethefollowingusingEITHERsubstitutionorelimination—yourchoice.Whicheveryouchoose,writeasentenceexplainingWHYyouselectedthemethodyoudid.
𝑦 = 2𝑥 − 1𝑥 + 3𝑦 = 9
2𝑥 − 5𝑦 = 94𝑥 + 2𝑦 = 3
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𝑦 = 𝑥 − 4𝑥 + 𝑦 = 4
𝑥 + 3𝑦 = 144𝑥 − 6𝑦 = 2
Solvethefollowingwordproblems.Bananashave105calorieseach.Appleshave75calorieseach.Inoneday,Maryonlyeatsthosetwofruits,andsheeats13piecesoffruit,andconsumesatotalof1215calories.HowmanyofeachfruitdidMaryconsume?Howdoyouthinkherstomachfeels?
OniTunes,youcandownloadsongsfor$1.29,andrentHDmoviesfor$5.99each.IfDominicspends$63.40for20totalpiecesofmedia(thisincludesbothmoviesandsongs),howmanymoviesdidherent?Howmanysongsdidhedownload?
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Solvethefollowingsystemsofequationsbygraphing,usingsubstitution,andusingelimination.
𝑦 = −32 𝑥 + 1
3𝑥 − 4𝑦 = 12
Solvebysubstitution:Solvebyelimination:
𝑥 − 𝑦 = −1𝑦 = 3𝑥 + 2
Solvebysubstitution:Solvebyelimination:
