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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
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Author:D.Tynan
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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
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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.
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
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Author:D.Tynan
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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.
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Author:D.Tynan
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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.
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
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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.
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Infinitesolutionsif
TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan