Module'3'Lesson'2:'Solving'Systems'of'Equations'Using'Matrices'
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In!Lesson!1,!we!learned!how!to!solve!a!system!of!linear!equations!by!graphing,!
substitution,!and!elimination.!!Here!in!Lesson!2!we!will!learn!a!new!way!to!solve!a!
system:!!matrices!via!a!Graphing!Calculator.!!Also,!we!will!work!with!multivariable'
equations.!!These!are!equations!that!have!three!variables,!namely!x,#y,#and!z.!!!
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Let’s!start!by!looking!at!a!matrix.!!A!“matrix”!(plural:!matrices)!is!a!rectangular!array!of!
numbers.!!Numbers!can!be!arranged!in!any!combination!of!rows!and!columns.!!The!
dimensions!of!a!matrix!are!denoted!as!!×!!where!! = !"#$%&!!"!!"#$!and!! =
!"#$%&!!"!!"#$%&'.!
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Ex.!!The!following!is!a!2!x!3!matrix.!!It!has!2!rows!and!3!columns.!
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3 −2 1
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0 5 −7
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In!order!to!solve!a!system!using!matrices,!we!will!use!our!Graphing!Calculator.!!More!
importantly,!we!will!be!using!the!MATRIX!command!(see!figure!below).!!!
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We!will!also!need!the!“RREF”!command!found!in!2nd!!!MATRIX!!Math.!!!
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RREF!stands!for!“Reduced!Row!Echelon!Form”.!!A!matrix!that!is!in!RREF!form!has!the!
following!characteristics:!
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• All!non!zero!rows!are!above!rows!of!all!zeros!
• The!leading!coefficient!of!a!non!zero!row!is!always!directly!to!the!right!of!the!
leading!coefficient!of!the!row!above!it!
• Every!leading!coefficient!is!1!and!it!is!the!only!non!zero!entry!in!its!column!
1 0 0
Ex.!! 0 1 0 !!This!is!an!example!of!a!matrix!in!reduced!row!echelon!form.!
0 0 1
2
3 −8
Ex.!! 0
1
4 !!This!is!an!example!of!a!matrix!not!in!reduced!row!echelon!
15 −6 0.5
form.!
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In!order!to!achieve!a!RREF!Matrix,!one!would!need!to!know!how!to!use!“Gaussian!
Elimination”,!often!calculated!by!hand.!!However,!we!will!not!be!learning!how!to!solve!a!
system!this!way!–!we!will!let!our!handy!Graphing!Calculator!do!the!work!for!us!!!Phew!!!
For!now,!you!just!need!to!remember!that!we!use!the!RREF!command!to!solve!a!system!
using!Matrices.!!"!
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Solving'Systems'of'Equations:''Two'Variables'
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Ex.!!Solve!the!following!system!using!matrices.!
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5! + 3! = 9
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2! − 4! = 14
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Note:!!Both!of!these!equations!are!in!standard!form!!" + !" = !.!!Linear!equations!
must!be!in!standard!form!to!use!the!matrix!feature!for!solving.!
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Step'1.''!Create!Matrix![A]!using!the!coefficients!and!constants!from!the!two!equations.!!
The!coefficients!are!the!numbers!paired!with!each!variable.!
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5! + 3! = 9
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2! − 4! = 14
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Click!2nd!!!MATRIX!and!scroll!over!to!EDIT.!
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Choose!Matrix![A].!!And!change!the!dimensions!to!2!x!3!(hitting!ENTER!after!each!entry)!
and!enter!in!the!numbers!as!they!appear!in!the!system.!!Enter!each!number!and!press!
ENTER!to!move!to!the!next!entry.!
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Step'2.''Click!2nd!!QUIT!to!return!to!the!Home!Screen.!!!Next,!press!2nd!!MATRIX!!!
MATH!B:!rref(.!
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Step'3.''Press!2nd!MATRIX!1:![A]!2!x!3.!!Press!ENTER.!
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You!only!need!Matrix![A].!!
Ignore!my!Matrix![B].!
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The!solution!matrix!is:!
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Recall!that!the!1st!column!contained!your!“x’s”!and!the!2nd!column!contained!your!“y’s”.!!
Therefore,!the!solution!is!x!=!3!and!y!=!`2,!or!more!specifically,!the!point!(3,!`2).!!!
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Consistent'&'Dependent'Systems'(Infinitely'Many'Solutions)'and'Inconsistent'
Systems'(No'Solutions)'
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If!in!fact!the!two!lines!are!the!same!line,!there!are!infinitely!many!solutions.!!If!this!is!the!
case,!you!will!receive!an!output!similar!to!the!following:!
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Notice'that'the'bottom'row'contains'all'zeros.!!If!the!bottom!row!is![0!!0!!0],!then!there!
are!infinitely'many'solutions!and!the!system!is!consistent!and!dependent.###
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If!the!two!lines!are!parallel,!there!will!be!no!solution.!!If!this!is!the!case,!you!will!receive!
an!output!similar!to!the!following:!
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Notice'that'the'bottom'row'contains'[0''0''1].!!If!the!bottom!row!has!a!0!for!x,!a!0!for!y,!
and!1!for!the!constant,!then!there!is!no!solution!and!the!system!is!inconsistent.#
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Solving'Systems'of'Linear'Equations:''Three'Variables'
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Ex.!!Solve!the!following!system!using!matrices.!!!
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! − 2! + 3! = 9
−! + 3! = −4 !
2! − 5! + 5! = 17
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Note:!!The!equations!are!written!in!standard!form!!" + !" + !" = !.!!!
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We!will!set!up!our!matrices!in!a!very!similar!manner!as!we!did!in!the!previous!example.!!
However,!our!dimensions!will!be!different!since!there!is!an!additional!variable,!z.!!!
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Step'1.!!Set!up!Matrix![A]!to!be!a!3!x!4!matrix.!!Enter!your!coefficients!and!constants.!!
You!will!enter!a!0!for!the!coefficient!for!z!in!the!second!equation.!!Also!notice!that!not!
all!columns!can!be!seen!at!the!same!time.!
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2nd!!!MATRIX!!!EDIT!!![A]!
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Step'2.!!2 !QUIT.!!2 !MATRIX!MATH!B:!rref(.!
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Step'3.''Press!2 !MATRIX!1:![A]!3!x!4.!!Press!ENTER.!
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Step'4.''Press!ENTER!to!find!the!solution.!
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Recall!that!your!“x’s”!are!in!column!1,!“y’s”!in!column!2,!and!“z’s”!in!column!3.!!The!
solution!to!the!system!is!the!point!(1,!`1,!2).!
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Note:!!The!same!rules!apply!for!infinitely#many#solutions!and!no#solution.!!
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Applications:'Word'Problems''
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Ex.!!Shondra!is!just!beginning!her!training!for!a!triathalon.!!In!her!training!routine!each!
week,!she!runs!6!times!more!than!she!swims,!and!she!bikes!4!times!more!than!she!runs.!!
One!week!she!trained!for!a!total!of!62!hours.!!How!far!did!she!run!that!week?!
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Let!! = !"#, ! = !"#$,!and!! = !"#$,!and!write!an!equation!for!each!relationship.!
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“She!runs!6!times!more!than!she!swims”:!!! = 6!!
“She!bikes!4!times!more!than!she!runs”:!!! = 4!!
“Trained!for!a!total!of!62!hours!“:!!! + ! + ! = 62!
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! = 6!
So!our!system!is!
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! = 4!
! + ! + ! = 62
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Rewrite!your!system!so!that!it!in!“standard!form”,!aligning!the!variables!r,#s,#and!b.!
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! − 6! = 0
−4! + ! = 0 !
! + ! + ! = 62
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Enter!your!system!into!Matrix![A]!being!careful!to!enter!zeros!where!necessary!and!
solve.!
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Therefore!she!runs!12!miles,!swims!2!miles,!and!bikes!48!miles.!!Answer:!She!runs!12!
miles.!
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Ex.''At!the!arcade,!Sam,!Trey,!and!Anna!played!video!racing!games,!pinball,!and!air!
hockey.!!Sam!spent!$6!total!for!6!racing!games,!2!pinball!games,!and!1!game!of!air!
hockey.!!Trey!spent!$12!total!for!3!racing!games,!4!pinball!games,!and!5!games!of!air!
hockey.!!Anna!spent!a!total!of!$12.25!for!2!racing!games,!7!pinball!games,!and!4!games!
of!air!hockey.!!How!much!did!each!of!the!games!cost?!
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Let!! = !"#$!!"!!"#$%&!!"#$%, ! = !"#$!!"#$%&'((!!"#$%,!and!
! = !"#$!!"!!"#!ℎ!"#$%!!"#$%.!
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Set!up!the!system:!!
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6! + 2! + ! = 6
3! + 4! + 5! = 12 !
2! + 7! + 4! = 12.25
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Enter!these!values!into!Matrix![A]!and!solve.!
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Therefore,!racing!games!cost!$0.50,!pinball!games!cost!$0.75,!and!air!hockey!games!cost!
$1.50.!
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