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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 18 "M
odule 1 : Algebra" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" 
{TEXT -1 21 "103 SOLVING EQUATIONS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 
252 "In this project we will use the following command packages. Type \+
and execute this line before begining the project below. If you re-ent
er the worksheet for this project, be sure to re-execute this statemen
t before jumping to any point in the worksheet." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 89 "_______________________________________________
__________________________________________" }}{PARA 4 "" 0 "" {TEXT 
-1 26 "A. Exact Solutions : Solve" }}{PARA 0 "" 0 "" {TEXT -1 89 "____
______________________________________________________________________
_______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 514 "In general, when we solv
e an equation, we usually obtain an exact solutions - which could be a
n integer, a fraction, or expressions with roots or  in them. Using a \+
calculator, we can convert exact solutions into decimal approximations
 of varying degrees of accuracy.  Using Maple, we can solve many types
 of equations and inequalities and obtain exact solutions. Lets look a
t some of the different types of equations and how their solutions app
ear. In a later section of this module well look at decimal solutions.
" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT -1 9 "EQUATIONS" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Here are some examples of a linear
 and quadratic equations.  equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( 4*x + 3 = 7);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "solve( 5* x^2 + 6*x =11 );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"\"#!#6\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "solve( 2*x^2 -x-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"
\"\"\"%F%*&F$F%-%%sqrtG6#\"#LF%F%,&F$F%*&#F%F&F%*$F(F%F%!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve( x^4 + 3*x^2 -4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&!\"\"\"\"\"^#\"\"#^#!\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( 5*x^2 + 6*x = 11);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$\"\"\"#!#6\"\"&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "solve( 2*x^2 - x - 4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,&#\"\"\"\"\"%F%*&F$F%-%%sqrtG6#\"#LF%F%,&F$F%*&#F%F&F%
*$F(F%F%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "Not all equations are solvable algebraically." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solv
e( x^5 - x + 1 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'-%'RootOfG6$,(*$)
%#_ZG\"\"&\"\"\"F+F)!\"\"F+F+/%&indexGF+-F$6$F&/F.\"\"#-F$6$F&/F.\"\"$
-F$6$F&/F.\"\"%-F$6$F&/F.F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 45 "Maple can also solve absolute value equations" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "abs( 3*x -5) = 17;   solve( %, x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$absG6#,&%\"xG\"\"$\"\"&!\"\"\"#<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$#\"#A\"\"$!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 11 "INEQALITIES" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 390 "In the examples above, Maple r
eturns one or more individual answers. However there are other types o
f problems, - for example those involving inequalities - where the ans
wer involves intervals instead of individual numbers. Happily, Maple a
djusts automatically to the type of problem involved. The same solve c
ommand will work on these problems. Maple expresses an interval as a r
eal range." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "solve( 4*x +3 < 7);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( 4*x +3 >=7 );" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$\"\"\"%)infinityG" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Here are some other \+
types of inequalities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 28 "-absolute value inequalities" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "abs( 139 - 41*x ) > 73;   solve(%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#2\"#t-%$absG6#,&!$R\"\"\"\"*&\"#TF*%\"xGF*F*" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$-%%OpenG6##\"$7#\"#T%)
infinityG-F$6$,$F,!\"\"-F'6##\"#mF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 25 "- polynomial inequalities" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(x^2 + 3 >= 7);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\"\"!\"#-F$6$\"\"#F
'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "-Hi
gher degree polynomial inequalities" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "poly := x^4 + 3*x^3 - 27*x^2 + 13*x + 42;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%polyG,,*$)%\"xG\"\"%\"\"\"F**&\"\"$F*)F(F
,F*F**&\"#FF*)F(\"\"#F*!\"\"*&\"#8F*F(F*F*\"#UF*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "solve( poly > 0,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#!\"(-F$6$-F*
6#F(-F*6#\"\"#-F$6$-F*6#\"\"$F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 61 "A1. Find the solutions to these equations and i
nequalities\n  " }}{PARA 0 "" 0 "" {TEXT -1 6 "  A. 3" }{XPPEDIT 19 1 
"x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 57 " - x - 4 = 0                    \+
                   \011\011\011B. " }{XPPEDIT 19 1 "x^2;" "6#*$%\"xG\"
\"#" }{TEXT -1 195 " -  x - 4  = 0                                    \+
  \011\011C. 19x + 31  7\n  D. | 5x+ 2 |  = 9           \011\011      \+
                         E. | 7 - 3x |   3                            \+
            \011\011F. 4" }{XPPEDIT 19 1 "x^2;" "6#*$%\"xG\"\"#" }
{TEXT -1 92 " + 3x  7  \n  G. | x3 | + | x2 | = | x+13 |              \+
       \011\011H.  | |x+1| -  |x-1| |  < 2" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 
"_____________________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 35 "B. Equations with
 Multiple Unknowns" }}{PARA 0 "" 0 "" {TEXT -1 89 "___________________
______________________________________________________________________
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 327 "The same
 solve command can be used to solve equations which have several unkno
wns. However, if there are multiple unknowns, you need to specify whic
h one you want Maple to solve for. You do this by including an additio
nal parameter in the command. The additional term, separated by a comm
a, is the variable you are solving for." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve( 3*x + 4*y = 17, \+
x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"yG#!\"%\"\"$#\"#<F'\"\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A = (a + b + c + d)/4; \+
  solve(%,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG,*%\"aG#\"\"\"
\"\"%*&F'F(%\"bGF(F(*&F'F(%\"cGF(F(*&F'F(%\"dGF(F(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,*%\"AG\"\"%%\"aG!\"\"%\"bGF'%\"dGF'" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "In case you forget t
he quadratic formula, just solve for it." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve( a*x^2 + b*x + \+
c = 0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&,&%\"bG!\"\"*$-%%sqr
tG6#,&*$)F&\"\"#\"\"\"F0*(\"\"%F0%\"aGF0%\"cGF0F'F0F0F0F3F'#F0F/,$*&,&
F&F'F(F'F0F3F'F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 163 "B1. Express each line equation into slope-intercept form (y= m
x+b) using the solve command.\n\011A. 6x  8y = 24      B. y - 3 = 2(x \+
+ 5)    C. x/6 + y/18 + 1/9 = 0  \n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 76 "B2. Given the formula F = (9/5)C + 32
, find a formula for C in terms of F.\n\n" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "B3. Solve each equation or formul
a for the indicated variable.\n\011A. 1/R = 1/R1 + 1/R2 + 1/R3   for R
\n\011B. 1/R = 1/R1 + 1/R2 + 1/R3   for R2\n\n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "B4. A formula can repres
ent a function. Its inverse is found by solving for x.\n\011A. y = 3x \+
- 7    \011\011B. y = (x-2)/(x+3)     \011C .y = (x-2)3 + 7  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 89 "____________________________________________________
_____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 20 "
C. Decimal Solutions" }}{PARA 0 "" 0 "" {TEXT -1 89 "_________________
______________________________________________________________________
__" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 310 "We saw how Maple can be used to solve eq
uations using the solve command. However, there are many cases where i
t is impossible to find an exact solution. Using the fsolve command we
 can get a numeric answer if one exists. Maple is able to do this by u
sing very sophisticated methods of approximating the answer." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Recall that we \+
could not solve this polynomial when we saw it above." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( x^
5 - x + 1 = 0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'-%'RootOfG6$,(*$)%#
_ZG\"\"&\"\"\"F+F)!\"\"F+F+/%&indexGF+-F$6$F&/F.\"\"#-F$6$F&/F.\"\"$-F
$6$F&/F.\"\"%-F$6$F&/F.F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 77 "However, using the fsolve command, we can now fin
d an answer in decimal form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve( x^5 - x +1 = 0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+yRIn6!\"*" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Sometimes there are multi
ple answers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "fsolve( x^4 = x^3 + 1, x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$!+M^s\">)!#5$\"+pvF!Q\"!\"*" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 
"_____________________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 23 "D. Solving The Lo
ng Way" }}{PARA 0 "" 0 "" {TEXT -1 89 "_______________________________
__________________________________________________________" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 284 "Although Maple can solve equations in a single step, you
 can also go through the same steps that you would do by hand. This is
 good for practicing the steps you need to take in solving a problem w
ithout going through all of the computations. Its also good for checki
ng your homework!!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 42 "equat := 4*(2*x + 1) - 11 = 2*x (x+1) - 3;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&equatG/,&%\"xG\"\")\"\"(!\"\",&-F
'6#,&F'\"\"\"F/F/\"\"#\"\"$F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 41 "This adds 7 to both sides of the equation
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "equat := % + ( 7=7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&equ
atG/,$%\"xG\"\"),&-F'6#,&F'\"\"\"F-F-\"\"#\"\"%F-" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "This subtracts 2x from bo
th sides" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "equat := % - (2*x = 2*x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&equatG/,$%\"xG\"\"',(-F'6#,&F'\"\"\"F-F-\"\"#\"\"%F-
*&F.F-F'F-!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 28 "This divides both sides by 6" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "equat := %/6;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&equatG/%\"xG,(-F&6#,&F&\"\"\"F+F+#F
+\"\"$#\"\"#F-F+*&#F+F-F+F&F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "D1.. Tr
y this problem by hand and verify the steps that Maple takes. Express \+
in English sentences what step is performed on each of the following l
ines (for example Add 39 to both sides.)           \n" }}{PARA 0 "" 0 
"" {TEXT -1 317 "\011> equat  :=  x/12  +  3  =  x/9  -  3;\n\011> equ
at  :=  %  *  lcm(12,9);                                              \+
               \n\011> equat  :=  %  -  (  3*x = 3*x ); \n\011> equat \+
 :=  %  + ( 108  = 108); \n\011> solve( equat, x );  \n\nD2. Find or c
reate a new problem and solve it step-by-step using Maple in this same
 way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "3 0" 1 }
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