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Solving with Matrices Student Worksheet 789101112 TI-Nspire Investigation Student 90min Introduction Simultaneouslinearequationscanbesolvedbyusingalgebraicapproachessuchastheelimination methodorsubstitutionmethod.Theycanalsobesolvedgraphicallybyfindingpointsofintersection. UsingCAStechnology,asolvecommandcanbeused. Anothermethodofsolvingsimultaneouslinearequationsrequirestheuseofmatrices.Thisapproach involvesrepresentingtheequationsasrowsofnumbersandthenmanipulatingthesetoreachsolutions. Part 1: Using matrices to solve a system of equations Considerthesimultaneousequations 5x–3y=22 x+4y=–14 Eachequationcanberepresentedbyarowofnumberswiththecoefficientsofxcomingfirst,followedby thecoefficientsofyandfinallytheconstantterms.Theserowsmakeupamatrixasshown. ⎡ 5 −3 22 ⎤ ⎢ ⎥ 1 4 −14 ⎦ ⎣ Inthismatrix,thefirstrowrepresentsthefirstequationandthesecondrowrepresentsthesecond equation.Althoughthisistheusualconvention,itisequallycorrecttorepresentthesesimultaneous equationsbythematrix ⎡ 1 4 −14 ⎤ ⎢ ⎥ 5 −3 22 ⎦ ⎣ Question1. Eachpairofsimultaneousequationsbelowcanberepresentedbyamatrix.Ineachcase,writedown thematrixthatcorrespondstothepairofequations. a) 3x+5y=21 6x–2y=6 b) 6x+5y=1 x–7y=8 © TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 2 SolvingwithMatrices–StudentWorksheet Onceamatrixrepresentingthesimultaneousequationshasbeencreateditispossibletotransformthis matrixtoobtainthesolutionsforxandy.Todothis,therrefcommandisused. Ithasalreadybeenshownthatsimultaneousequations5x–3y=22andx+4y=–14canberepresented ⎡ ⎤ bythematrix ⎢ 5 −3 22 ⎥ 1 4 −14 ⎦ ⎣ Thescreenshotshowshowtherrefcommandisusedtotransformthematrix. Thetransformedmatrixrepresentstheequations 1x+0y=2 0x+1y=–4 Thereforex=2andy=–4isthesimultaneoussolutiontotheseequations. Thissolutioncanbecheckedusingsubstitution. • • 5x–3y=5×2–3×–4=10+12=22,so5x–3y=22 x+4y=2+4×–4=2–16=–14,sox+4y=–14 Question2. Ineachofthefollowing,therrefcommandhasbeenusedtosolvesimultaneousequations.Ineach case,writedowntheequationsbeingsolvedandtheirsimultaneoussolution. Equations:4x+2y=10andx+3y=10 Solution:x=1andy=3 © Equations:3x–y=5and2x+6y=–15 Solution:x=3/4andy=−11/4 TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 3 SolvingwithMatrices–StudentWorksheet Part 2: Using the ‘rref’ command Nowconsiderthefollowingsimultaneousequations. x–2y=10 2x+4y=16 Createamatrixwithrowsthatarethecoefficientsofxandyandtheconstanttermsintheequations. ⎡ 1 −2 10 ⎤ ⎢ ⎥ 2 4 16 ⎦ ⎣ ToenterthismatrixontheTI-NspireCAS: • PressHOME-1tocreateanewdocument,andthenpress1toaddaCalculatorpage. • PresstheTemplateskey(seescreenabove). • Selectthe‘CreateaMatrix’icon (itlookslikea3by3matrix–seescreenabove) • Inthedialogboxthatfollows – Forthenumberofrows,type2. – Forthenumberofrows,type3. • PressENTERtocreatethe2by3matrixtemplate • Typeintherequiredvalues,thenpressENTER. ⎛⎡ ⎤⎞ Toapplytherrefcommand,type rref ⎜ ⎢ 1 −2 10 ⎥⎟ ⎝ ⎣ 2 4 16 ⎦⎠ [Note:TherrefcommandcanalsobeaccessedviatheCatalog,orjustbytypinginthelettersdirectly.] Theresultantscreenshowsthattherrefcommandhastransformedtheoriginalpairofequationsinto 1x+0y=9 0x+1y=−1/2 Sothesolutionisx=9andy=−1/2. © TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 4 SolvingwithMatrices–StudentWorksheet Question3. Usetherrefcommandtosolveeachsetofsimultaneousequationsbelow. a) 3x+5y=21and6x–2y=6 x=2,y=3 b) 6x+5y=1andx–7y=8 x=1,y=–1 c) x+3y=–2and2x–5y=18 x=4,y=–2 d) 4x+3y=3and2x–6y=–11 x=1/2,y=1/3 Thesamemethodcanbeusedtosolvelargersystemsoflinearequations. Question4. Usetherrefcommandtofindvaluessimultaneousequationsofthepronumeralsthatsatisfyeachof thefollowingsystemsofequations a) 3x+2y–z=5 4x+3y+z=8 2x+2y–z=4 x=1,y=6/5,z=2/5 b) 2x+3y+2z=10 –x+4y+z=6 4x–2y+z=12 x=42,y=34,z=–88 Question5. Whatrelationshipexistsbetweenthenumberofequationsandthenumberofunknowns? Thenumberofequationsisequaltothenumberofunknowns. Part 3: How many solutions are possible? Sofar,allofthesystemsofequationshaveasinglesolution.Butthisisnottrueforallsuchsystems.In thispartoftheexploration,welookatthedifferenttypesofsolutions,andhowtherrefcommand handleseachsituation.Tobegin,considerthefollowingsystemofequations. x+y=1 x–y=3 Usingtherrefcommand,thissystemhasasolutionat x=2andy=–1(seescreen). Graphically,wecanrepresentthissolutionasthepointat whichthetwoassociatedlinesintersect. Ifwerewritethetwoequationsintheformy=mx+c,weget thefollowing. x+y=1⇒y=–x+1 x–y=3⇒y=x–3 Ifwegraphthesetwoequations,wewouldfindthatthey intersectatthepoint(2,–1)asthescreenshows.Thisisan exampleofasystemofequationsthathasauniquesolution. © TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 5 SolvingwithMatrices–StudentWorksheet Nowconsideranothersystemofequations. x+y=1 x+y=3 Usingtherrefcommand,thecalculatorreportsthatthis systemhasa‘solution’atx+y=0and0=1 Astheequation0=1isfalse,wecansaythatthissystem hasnosolution. Graphically,thesolutiontoapairoflinearsimultaneous equationsisthepointatwhichthetwoassociatedlines intersect.Ifwerewritethetwoequationsintheform y=mx+c,wegetthefollowing. x+y=1⇒y=–x+1and x+y=3⇒y=–x+3 Bygraphingthesetwoequations,itcanbeobservedthat thelinesareparallel(thegradientsareboth–1),sothey neverintersect!Thisisanexampleofasystemof equationsthathasnosolution. Finallyconsiderthefollowingsystemofequations. x+y=1 2x+2y=2 Usingtherrefcommand,thissystemhasasolutionat x+y=1and0=0 Thestatement0=0istrueandthereexistsaninfinitesetofsolutionsthatsatisfytheequationx+y=1, someofwhichinclude,x=0andy=1,x=1andy=0,x=–2andy=3,x=0.5andy=0.5.Thisiswhythe rrefcommandhasbeenunabletoreducethesystemtoauniquesolution.Insteaditisreportingthat thereismorethanonesolution,infactanypairofxandyvaluesthataddto1! Graphically,wecanrepresentthissolutionasanypointatwhichthetwoassociatedlinesintersect.That is,ifwerewritethetwoequationsintheformy=mx+c, wegetthefollowing. x+y=1⇒y=–x+1 2x+2y=2⇒2y=–2x+2⇒y=1/2(−2x+2) Ifwegraphthesetwoequations,wewouldfindthatthe graphsareidentical(theyhavethesamegraph),asthe screenshows.Thisisanexampleofasystemofequations thathasinfinitesolutions. © TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan 6 SolvingwithMatrices–StudentWorksheet Question6. Forthefinalcase(infinitesolutions),explainhowthesecondequationisrelatedtothefirstequation. Thesecondequationisthesameequationasthefirstequation,justinanotherform.Thatis,itisan equivalentequation,madedifferentbyeachterminthesecondequationbeingdoublethe correspondingterminthefirstequation. Question7. Forthefollowingsystemsofequations,usetherrefcommandtodeterminewhethertheyhavea uniquesolution,nosolutions,orinfinitesolutions.Ifthereisauniquesolution,givethissolution.If thereareinfinitesolutions,givetheequationthatdescribesallthevaluesofxandythatwould providesolutions. a) x+y=1 –x+2y=3 Uniquesolution;x=0,y=1 b) x+y=1 –x–y=2 Nosolution c) 2x+y=1 x+2y=3 Uniquesolution;x=–1/3;y=5/3 d) 2x+2y=2 3x+3y=3 Infinitesolutions;x+y=1 Challenge Determinethevaluesofmandnforwhichthesystem willhave 3x+3y=6 mx+y=n a) auniquesolution Uniquesolutionif b) nosolutions Nosolutionif c) infinitesolutions. © Infinitesolutionsif TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes providedallacknowledgementsassociatedwiththismaterialaremaintained. Author:D.Tynan