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{SECT 0 {PARA 18 "" 0 "" {TEXT 256 35 "SOLVING SYSTEMS OF LINEAR EQUAT
IONS" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 28 
"Linear Algebra Maple Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 10 "Objective:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "To solve
 various systems of equations; to understand when a system has zero, o
ne or infinitely many solutions." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
0 "" }{TEXT 259 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 43 "a) Solve the following system of equatio
ns:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 16 "
3x - 2y + z = 12" }}{PARA 257 "" 0 "" {TEXT -1 16 "2x +  y -5z = -1" }
}{PARA 258 "" 0 "" {TEXT -1 16 "x -  y - 8z  = 4" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "b) Show a plot of the geo
metric interpretation of your solution." }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 9 "Solution:" }{TEXT 262 0 "
" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 282 2 "a)" }{TEXT -1 1 " " }{TEXT 
281 19 "Algebraic Solution:" }}{PARA 0 "" 0 "" {TEXT -1 71 "Maple can \+
solve a system of equations in several variables as follows: " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "solve(\{3*x-2*y+z=12,2*x+y-5
*z=-1,x-y-8*z=4\},\{x,y,z\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/
%\"xG#\"$.\"\"#k/%\"zG#\"\"*F(/%\"yG#!$D#F(" }}}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 283 29 "
 b) Geometric interpretation:" }}{PARA 0 "" 0 "" {TEXT -1 65 "We activ
ate the Maple linear algebra, plots and Lamp*  packages: " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "with (linalg);: with (plots): with \+
(Lamp);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSc
hmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%'addro
wG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%%bandG%&basisG%'bezoutG
%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG%)colspac
eG%(colspanG%*companionG%'concatG%%condG%)copyintoG%*crossprodG%%curlG
%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG%*eige
nvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&equalG
%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG%*froben
iusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(
hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG%*inne
rprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobianG%'
jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'matrixG%
&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG%*nul
lspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+randvec
torG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scal
armulG%-singularvalsG%&smithG%,stackmatrixG%*submatrixG%*subvectorG%)s
umbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*transpose
G%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#7_p%'Axis3dG%*BackidmatG%*BacksolveG%(Bandma
tG%*BasisgridG%$BugG%&Bug3dG%'Bug3dhG%)CharpolyG%+Chevron3dhG%&ClockG%
*ColorlistG%.ComponentplotG%)CopyintoG%*CrossprodG%'Cube3dG%(Cube3dhG%
$DetG%(DiagmatG%(DotprodG%*DrawlinesG%+DrawmatrixG%+DrawplanesG%(Drawv
ecG%*Drawvec3dG%&EvalmG%(EvaluesG%)EvectorsG%'ExpandG%*GenmatrixG%)Get
colorG%%GridG%)GridgameG%*GridvectsG%)HeadtailG%(Hotel3dG%)Hotel3dhG%&
HouseG%)House3dhG%&IdmatG%(InverseG%&Jet3dG%'Jet3dhG%*JordanmatG%(Lamp
3dhG%&LamphG%(LetterLG%(LetterNG%(LetterSG%(LetterXG%$MagG%)MapcolorG%
,MappicturesG%/MatrixtovectorG%)MatsolveG%&MovieG%*NullbasisG%%PathG%(
ProjectG%+ProjectmatG%-Projectmat3dG%)QuestionG%(RandmatG%'ReduceG%+Re
flectmatG%-Reflectmat3dG%*ResidualsG%*RotatematG%,Rotatemat3dG%&RowopG
%+SymbollistG%+TrajectoryG%-Trajectory3dG%*TransformG%-TranslatematG%/
Translatemat3dG%)UnitspanG%*VandermatG%+VectorgridG%+VectorlineG%-Vect
ranslateG%*XshearmatG%*YshearmatG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 82 "Each equation in three variables defines \+
a plane.  Let us call them eq1, eq2, eq3." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "eq1 := 3*x-2*y+z=12; eq2 := 2*x+y-5*z=-1; eq3 := x-y-
8*z=4;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,(*&\"\"$\"\"\"%\"
xGF)F)*&\"\"#F)%\"yGF)!\"\"%\"zGF)\"#7" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$eq2G/,(*&\"\"#\"\"\"%\"xGF)F)%\"yGF)*&\"\"&F)%\"zGF)!\"\"F/" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,(%\"xG\"\"\"%\"yG!\"\"*&\"
\")F(%\"zGF(F*\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 105 "Solving a system
 of equations means finding a point of intersection of the planes the \+
equations define.  " }}{PARA 0 "" 0 "" {TEXT -1 104 "The common point \+
may or may not always exist.  We can see this clearly by plotting the \+
graphs of planes." }}{PARA 0 "" 0 "" {TEXT -1 108 "To investigate the \+
geometric situation further, we click on the graph and get a graphics \+
menubar as follows." }}{PARA 0 "" 0 "" {OLE 1 80395 1 "[xm]Br=WfoRrB::
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bdnFbdn" 1 2 0 1 10 0 2 1 1 2 1 1.000000 61.000000 42.000000 0 0 "Curv
e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}}{PARA 0 "" 0 "
" {TEXT -1 77 "_______________________________________________________
______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "
" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 10 "ASSIGNMENT" }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 10 "Problem 1:" }
{TEXT 270 0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }{TEXT 273 0 "" }{TEXT 
274 0 "" }{TEXT 275 0 "" }{TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 93 "a) Why does this system have no solu
tion? What happens when you try to use Maple to solve it?" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "                  \+
                              x + 2y = 5" }}{PARA 0 "" 0 "" {TEXT -1 
58 "                                              2x + 4y = 20" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "b) Use th
e \"drawlines\" command to see graphically what is happening." }}
{PARA 3 "" 0 "" {TEXT 266 1 " " }{TEXT 267 0 "" }{TEXT 268 0 "" }
{TEXT 269 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 277 10 
"Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 98 "a) What if three planes all have a line in common? How many sol
utions will there be to the system?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 124 "b) Solve the following system and graph
 the planes represented by the equations. Interpret the     solution M
aple gives you." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 63 "                                                 x + y - \+
z  = 2" }}{PARA 0 "" 0 "" {TEXT -1 63 "                               \+
                2x + y - z  = -3" }}{PARA 0 "" 0 "" {TEXT -1 61 "     \+
                                          3x +2y-2z = -1" }}{PARA 0 "
" 0 "" {TEXT -1 42 "c) What do you notice about the equations?" }}
{PARA 0 "" 0 "" {TEXT -1 90 "                                         \+
                                                 " }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 0 "" }{TEXT 278 20 "Problem 3 (Summary):" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "a) Describe the co
nditions under which a system of equations in " }{TEXT 279 3 "two" }
{TEXT -1 117 " variables will have zero, one or  infinitely many solut
ions (there can be one, two or more equations in the system)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "b) (Chall
enge). Answer the above question for a system of equations in " }
{TEXT 280 5 "three" }{TEXT -1 11 " variables." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 77 "________________________
_____________________________________________________" }}{PARA 0 "" 0 
"" {TEXT -1 124 "MSIP Grant #P120A80089-98: \"Three Urban Calculus Ref
orm Programs: Adopting the Best\"