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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 52 "Steady State Material Bal
ances on a Separation Train" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "We begin by creating a list of \+
component identities. We could use numbers, chemical formulae, the com
ponent names; here we use the first letter of the component names." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "components:=[X,S,T,B];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+componentsG7&%\"XG%\"SG%\"TG%\"BG" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Next, we create a list of unit
s.  This problem has three distillation columns which shall be identif
ied as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Units:=[
Col1,Col2, Col3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&UnitsG7%%%Col1
G%%Col2G%%Col3G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "The streams w
ill also have to be identified.  Again, we could use any indexing meth
od that is convenient.  Here we number the streams" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 25 "Streams:=[seq(i,i=1..7)];" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%(StreamsG7)\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The number of streams is" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "NumStreams := nops(Streams);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+NumStreamsG\"\"(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 243 "The next step is to set up the material \+
balances.  To help us do this in a systematic way we create lists of t
he input streams to and output streams leaving each process unit.  The
 streams entering and leaving the first distillation column are" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Inputs[Col1]:=[1]; Outputs[C
ol1]:=[2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'InputsG6#%%Col1G7#
\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(OutputsG6#%%Col1G7$\"\"#
\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The streams entering and
 leaving the second column are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 40 "Inputs[Col2]:=[2]; Outputs[Col2]:=[4,5];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'InputsG6#%%Col2G7#\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%(OutputsG6#%%Col2G7$\"\"%\"\"&" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 57 "and the streams entering and leaving the third col
umn are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Inputs[Col3]:=[3
]; Outputs[Col3]:=[6,7];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Inputs
G6#%%Col3G7#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(OutputsG6#%%C
ol3G7$\"\"'\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "The component
 material balances for all units can now be created automatically as f
ollows:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "j:='j': i:='i':\nfor u \+
in Units do\n  lprint(`Material Balances for `,u);\n  for i in compone
nts do\n    CMB[i,u] := add( F[j] * x[i,j], j = Inputs[u]) = add( F[j]
 * x[i,j], j = Outputs[u]);\n    print(CMB[i,u]);\nod; od;" }}{PARA 6 
"" 1 "" {TEXT -1 29 "Material Balances for    Col1" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&&%\"FG6#\"\"\"F(&%\"xG6$%\"XGF(F(,&*&&F&6#\"\"#F(&
F*6$F,F1F(F(*&&F&6#\"\"$F(&F*6$F,F7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&&%\"FG6#\"\"\"F(&%\"xG6$%\"SGF(F(,&*&&F&6#\"\"#F(&F*6$F,F1F(F
(*&&F&6#\"\"$F(&F*6$F,F7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%
\"FG6#\"\"\"F(&%\"xG6$%\"TGF(F(,&*&&F&6#\"\"#F(&F*6$F,F1F(F(*&&F&6#\"
\"$F(&F*6$F,F7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"FG6#\"\"
\"F(&%\"xG6$%\"BGF(F(,&*&&F&6#\"\"#F(&F*6$F,F1F(F(*&&F&6#\"\"$F(&F*6$F
,F7F(F(" }}{PARA 6 "" 1 "" {TEXT -1 29 "Material Balances for    Col2
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"FG6#\"\"#\"\"\"&%\"xG6$%\"X
GF(F),&*&&F&6#\"\"%F)&F+6$F-F2F)F)*&&F&6#\"\"&F)&F+6$F-F8F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"FG6#\"\"#\"\"\"&%\"xG6$%\"SGF(F
),&*&&F&6#\"\"%F)&F+6$F-F2F)F)*&&F&6#\"\"&F)&F+6$F-F8F)F)" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/*&&%\"FG6#\"\"#\"\"\"&%\"xG6$%\"TGF(F),&*&&F&6
#\"\"%F)&F+6$F-F2F)F)*&&F&6#\"\"&F)&F+6$F-F8F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%\"FG6#\"\"#\"\"\"&%\"xG6$%\"BGF(F),&*&&F&6#\"\"%F)
&F+6$F-F2F)F)*&&F&6#\"\"&F)&F+6$F-F8F)F)" }}{PARA 6 "" 1 "" {TEXT -1 
29 "Material Balances for    Col3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*&&%\"FG6#\"\"$\"\"\"&%\"xG6$%\"XGF(F),&*&&F&6#\"\"'F)&F+6$F-F2F)F)*&&
F&6#\"\"(F)&F+6$F-F8F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"FG6
#\"\"$\"\"\"&%\"xG6$%\"SGF(F),&*&&F&6#\"\"'F)&F+6$F-F2F)F)*&&F&6#\"\"(
F)&F+6$F-F8F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"FG6#\"\"$\"
\"\"&%\"xG6$%\"TGF(F),&*&&F&6#\"\"'F)&F+6$F-F2F)F)*&&F&6#\"\"(F)&F+6$F
-F8F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"FG6#\"\"$\"\"\"&%\"x
G6$%\"BGF(F),&*&&F&6#\"\"'F)&F+6$F-F2F)F)*&&F&6#\"\"(F)&F+6$F-F8F)F)" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "In the above Maple constructio
n we have created the component material balances for each component (
the inner loop) and for each process unit (the outer loop).\n\nThe tot
al molar flows are given the symbol " }{TEXT 256 1 "F" }{TEXT -1 5 " a
nd " }{TEXT 257 1 "x" }{TEXT -1 404 " refers to the mole fraction of  \+
some component.  The first index of the component mole fraction identi
fies the component in question, the second associates that quantity wi
th a particular process stream.  This double loop will work with all s
imple material balances regardless of complexity provided we have iden
tified the components, the units and the inputand output streams assoc
iated with each unit." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The mole fraction summation equati
ons can be created as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 112 "SumEqn:='SumEqn': i:='i':\nfor j in Streams do\n   SumEqn[j]:
=add(x[i,j],i=components)=1;\n   print(SumEqn[j]);\nod:" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/,*&%\"xG6$%\"XG\"\"\"F)&F&6$%\"SGF)F)&F&6$%\"TG
F)F)&F&6$%\"BGF)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6$%\"
XG\"\"#\"\"\"&F&6$%\"SGF)F*&F&6$%\"TGF)F*&F&6$%\"BGF)F*F*" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/,*&%\"xG6$%\"XG\"\"$\"\"\"&F&6$%\"SGF)F*&F&6$%
\"TGF)F*&F&6$%\"BGF)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6
$%\"XG\"\"%\"\"\"&F&6$%\"SGF)F*&F&6$%\"TGF)F*&F&6$%\"BGF)F*F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6$%\"XG\"\"&\"\"\"&F&6$%\"SGF
)F*&F&6$%\"TGF)F*&F&6$%\"BGF)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
,*&%\"xG6$%\"XG\"\"'\"\"\"&F&6$%\"SGF)F*&F&6$%\"TGF)F*&F&6$%\"BGF)F*F*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6$%\"XG\"\"(\"\"\"&F&6$%
\"SGF)F*&F&6$%\"TGF)F*&F&6$%\"BGF)F*F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 239 "The component material balances and the mole fraction su
mmation equations comprise the complete set of independent equations f
or this kind of problem.  Any other balance equation can be created fr
om simple combinations of these equations. " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "We now create a
 set independent equations to describe the flowsheet " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Eqns:=\{seq(seq(CMB[i,u],i=componen
ts),u=Units),seq(SumEqn[j],j=Streams)\};" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%%EqnsG<5/,*&%\"xG6$%\"XG\"\"'\"\"\"&F)6$%\"SGF,F-&F)6$%\"TGF,F
-&F)6$%\"BGF,F-F-/*&&%\"FG6#\"\"$F-&F)6$F3F<F-,&*&&F:6#F,F-F1F-F-*&&F:
6#\"\"(F-&F)6$F3FFF-F-/,*&F)6$F+FFF-&F)6$F0FFF-FGF-&F)6$F6FFF-F-/*&F9F
-&F)6$F6F<F-,&*&FAF-F4F-F-*&FDF-FOF-F-/*&&F:6#F-F-&F)6$F+F-F-,&*&&F:6#
\"\"#F-&F)6$F+F\\oF-F-*&F9F-&F)6$F+F<F-F-/*&FZF-&F)6$F0F-F-,&*&FjnF-&F
)6$F0F\\oF-F-*&F9F-&F)6$F0F<F-F-/*&FZF-&F)6$F6F-F-,&*&FjnF-&F)6$F6F\\o
F-F-FRF-/*&FZF-&F)6$F3F-F-,&*&FjnF-&F)6$F3F\\oF-F-F8F-/Fin,&*&&F:6#\"
\"%F-&F)6$F+FbqF-F-*&&F:6#\"\"&F-&F)6$F+FhqF-F-/Fgo,&*&F`qF-&F)6$F0Fbq
F-F-*&FfqF-&F)6$F0FhqF-F-/,*FfnF-FdoF-FgpF-F_pF-F-/Fjp,&*&F`qF-&F)6$F3
FbqF-F-*&FfqF-&F)6$F3FhqF-F-/,*F]oF-FhoF-F[qF-FcpF-F-/,*F`oF-F[pF-F=F-
FSF-F-/Fbp,&*&F`qF-&F)6$F6FbqF-F-*&FfqF-&F)6$F6FhqF-F-/,*FcqF-F^rF-Fhr
F-FdsF-F-/F_o,&*&FAF-F(F-F-*&FDF-FKF-F-/Fjo,&*&FAF-F.F-F-*&FDF-FMF-F-/
,*FiqF-FarF-F[sF-FgsF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The l
ist of variables appearing in these equations is easily found" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Vars := indets(Eqns,name);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%VarsG<E&%\"xG6$%\"SG\"\"'&F'6$%\"
TG\"\"(&F'6$F-F*&F'6$F)F.&F'6$%\"XGF.&F'6$F5F*&F'6$%\"BG\"\"&&F'6$F:\"
\"%&F'6$F-F;&F'6$F-F>&F'6$F:F*&F'6$F:F.&F'6$F5\"\"$&F'6$F5\"\"#&F'6$F5
\"\"\"&F'6$F)FO&%\"FG6#FL&FS6#FO&FS6#FI&F'6$F)FI&F'6$F)FL&F'6$F:FI&F'6
$F:FL&F'6$F:FO&F'6$F-FI&F'6$F-FL&F'6$F-FO&FS6#F>&FS6#F;&F'6$F5F;&F'6$F
5F>&F'6$F)F>&FS6#F.&FS6#F*&F'6$F)F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "and we can count them using the nops function:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Nvars := nops(\");" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&NvarsG\"#N" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
52 "The number of equations is computed in a similar way" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Neqns:=nops(Eqns);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&NeqnsG\"#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The number of degrees of freedom i
s the difference between these two numbers" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "DegFree:=Nvars-Neqns;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(DegFreeG\"#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "This is
 the number of  variables that we must specify before we have a consis
tent set of equations that can be solved (numerically, at least)." }}
{PARA 0 "" 0 "" {TEXT -1 117 "\nWe know the flow and composition of th
e feed stream (note that only three of the feed mole fractions are spe
cified)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Specs[1]:=F[1]=
70;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"\"\"/&%\"FGF&\"#q
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Specs[2]:=x[X,1]=0.15; \+
Specs[3]:=x[S,1]=0.25; Specs[4]:=x[T,1]=0.4;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&SpecsG6#\"\"#/&%\"xG6$%\"XG\"\"\"$\"#:!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"\"$/&%\"xG6$%\"SG\"\"\"$
\"#D!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"\"%/&%\"xG6$
%\"TG\"\"\"$F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "We need t
o specify another 12 variables and must look to the problem statement \+
to find out what other specifications we may make." }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 272 "In the p
roblem as posed the composition of all final product streams is given.
 However, in view of the mole fraction summation equations we may only
 specify 3 from each stream; the choice of which to omit is arbitrary \+
- we leave out the lowest mole fraction in most cases." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "Specs[5]:=x[X,4]=0.07; Specs[6]:=x[B,4]=0.35; \+
Specs[7] := x[T,4]=0.54;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG
6#\"\"&/&%\"xG6$%\"XG\"\"%$\"\"(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%&SpecsG6#\"\"'/&%\"xG6$%\"BG\"\"%$\"#N!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&SpecsG6#\"\"(/&%\"xG6$%\"TG\"\"%$\"#a!\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Specs[8]:=x[X,5]=0.18; Specs
[9]:=x[B,5]=0.16; Specs[10] := x[T,5]=0.42;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&SpecsG6#\"\")/&%\"xG6$%\"XG\"\"&$\"#=!\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"\"*/&%\"xG6$%\"BG\"\"&$\"#;!\"
#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"#5/&%\"xG6$%\"TG\"
\"&$\"#U!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Specs[11]:=
x[X,6]=0.15; Specs[12]:=x[B,6]=0.21; Specs[13] := x[T,6]=0.54;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"#6/&%\"xG6$%\"XG\"\"'$\"
#:!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"#7/&%\"xG6$%\"
BG\"\"'$\"#@!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"#8/&
%\"xG6$%\"TG\"\"'$\"#a!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
73 "Specs[14]:=x[X,7]=0.24; Specs[15]:=x[S,7]=0.65; Specs[16] := x[T,7
]=0.10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"#9/&%\"xG6$%
\"XG\"\"($\"#C!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"#:
/&%\"xG6$%\"SG\"\"($\"#l!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&Sp
ecsG6#\"#;/&%\"xG6$%\"TG\"\"($\"#5!\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 81 "We now have the right number of specification equations w
hich we combine in a set" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"SpecEqns := \{seq(Specs[l],l=1..DegFree)\};" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%)SpecEqnsG<2/&%\"xG6$%\"BG\"\"'$\"#@!\"#/&F(6$%\"XGF+
$\"#:F./&F(6$F*\"\"&$\"#;F./&F(6$%\"TGF8$\"#UF./&F(6$F>F+$\"#aF./&F(6$
%\"SG\"\"($\"#lF./&F(6$F2FJ$\"#CF./&%\"FG6#\"\"\"\"#q/&F(6$F>FJ$\"#5F.
/&F(6$F2FVF3/&F(6$FIFV$\"#DF./&F(6$F>FV$\"\"%!\"\"/&F(6$F2Fco$FJF./&F(
6$F>FcoFD/&F(6$F*Fco$\"#NF./&F(6$F2F8$\"#=F." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "nops(SpecEqns);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We augment the set of \+
equations with the specification equations" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "AllEqns := Eqns union SpecEqns;" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%(AllEqnsG<E/&%\"xG6$%\"BG\"\"'$\"#@!\"#/,*&F(6$%\"X
GF+\"\"\"&F(6$%\"SGF+F4&F(6$%\"TGF+F4F'F4F4/F1$\"#:F./&F(6$F*\"\"&$\"#
;F./&F(6$F:FA$\"#UF./*&&%\"FG6#\"\"$F4&F(6$F:FNF4,&*&&FL6#F+F4F8F4F4*&
&FL6#\"\"(F4&F(6$F:FXF4F4/F8$\"#aF./&F(6$F7FX$\"#lF./,*&F(6$F3FXF4FinF
4FYF4&F(6$F*FXF4F4/F_o$\"#CF./&FL6#F4\"#q/*&FKF4&F(6$F*FNF4,&*&FSF4F'F
4F4*&FVF4FaoF4F4/FY$\"#5F./&F(6$F3F4F</*&FgoF4FepF4,&*&&FL6#\"\"#F4&F(
6$F3F]qF4F4*&FKF4&F(6$F3FNF4F4/&F(6$F7F4$\"#DF./*&FgoF4FdqF4,&*&F[qF4&
F(6$F7F]qF4F4*&FKF4&F(6$F7FNF4F4/*&FgoF4&F(6$F*F4F4,&*&F[qF4&F(6$F*F]q
F4F4F[pF4/*&FgoF4&F(6$F:F4F4,&*&F[qF4&F(6$F:F]qF4F4FJF4/F[s$\"\"%!\"\"
/Fjp,&*&&FL6#FcsF4&F(6$F3FcsF4F4*&&FL6#FAF4&F(6$F3FAF4F4/F[r,&*&FhsF4&
F(6$F7FcsF4F4*&F]tF4&F(6$F7FAF4F4/Fjs$FXF./,*FepF4FdqF4F[sF4FcrF4F4/&F
(6$F:FcsFfn/&F(6$F*Fcs$\"#NF./F^s,&*&FhsF4F^uF4F4*&F]tF4FEF4F4/,*F^qF4
F\\rF4F_sF4FgrF4F4/,*FaqF4F_rF4FOF4F\\pF4F4/Ffr,&*&FhsF4FauF4F4*&F]tF4
F?F4F4/,*FjsF4FdtF4F^uF4FauF4F4/F`q,&*&FSF4F1F4F4*&FVF4F_oF4F4/F^r,&*&
FSF4F5F4F4*&FVF4FinF4F4/,*F_tF4FgtF4FEF4F?F4F4/F_t$\"#=F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 71 "We check to see if the numbers of equatio
ns and variables are the same." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "nops(AllEqns); nops(Vars);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#N" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 44 "Now we can ask Maple to solve the equations." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "result:=solve(AllEqns,Vars);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'resultG<E/&%\"xG6$%\"SG\"\"%$\"
+++++S!#6/&F(6$%\"XG\"\"$$\"+++++@!#5/&%\"FG6#\"\"'$\"++++]()!\"*/&F96
#\"\"&$\"++++]<!\")/&F(6$%\"BG\"\"($\"+++++5F./&F(6$FI\"\"\"$\"+++++?F
6/&F(6$F2\"\"#$\"++++S6F6/&F96#FJFC/&F(6$F*F;$FLF6/&F96#F+$\"++++DEFE/
&F(6$F*FB$\"+++++CF6/&F(6$FIF;F4/&F(6$%\"TGFB$\"+++++UF6/&F(6$FIFB$\"+
++++;F6/&F(6$F2F;$\"+++++:F6/&F(6$FjoF;$\"+++++aF6/&F(6$F*FJ$\"+++++lF
6/&F(6$F2FB$\"+++++=F6/&F(6$FjoFP$F-F6/&F(6$F2F+$\"+++++qF./&F(6$FjoF+
Fjp/&F(6$FIF+$\"+++++NF6/&F(6$F2FJFbo/&F(6$FjoFJFin/&F(6$F*FP$\"+++++D
F6/&F(6$F2FPFep/&F(6$FIFV$\"++++SFF6/&F(6$F*FV$\"+++++7F6/&F(6$F*F3$\"
+nmmmYF6/&F(6$FjoF3$\"+nmmmCF6/&F(6$FIF3$\"+nmmmwF./&F(6$FjoFV$\"++++?
\\F6/&F96#F3F]o/&F96#FV$\"++++vVFE/&F96#FP$\"#q\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 68 "We check the solution by substituting the resul
t into the equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "su
bs(result,Eqns);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<//$\"+++++5!\"*\"
\"\"/$\"+,++vkF'$\"++++vkF'/$\"+++]7?F'F//$\"++++]5!\")F2/$\"++++]<F4F
6/$\"+++++9F4F9/$\"+++++GF4F</$\"+++]()\\F'F?/$\"++++]_F'FB/$\"+++]_@F
4FE/$\"+++v)>\"F4FH/$\"+++]7bF'FK/$\"++++D7F4FN" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 81 "and we see that the result that Maple computed does \+
indeed satisfy the equations." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 
1 1803 }