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{SECT 0 {SECT 0 {PARA 256 "" 0 "" {TEXT 256 28 "Section 4: Solving Equ
ations" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "In this section
you will learn how to apply Maple's " }{TEXT 276 8 "solve( )" }{TEXT
-1 21 " command to find the " }{TEXT 275 5 "exact" }{TEXT -1 197 " sol
utions of equations (when this is possible). You may recall from Preca
lculus that we are not able in many cases to find exact solutions to e
quations and so we rely on numerical solvers to find " }{TEXT 278 11 "
approximate" }{TEXT -1 55 " solutions. Later in this section you will \+
use Maple's " }{TEXT 277 10 "fsolve( ) " }{TEXT -1 121 "command to fin
d decimal approximations for solutions. The solution to linear systems
of equations will also be discussed." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12
"with(plots):" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Entering and Ma
nipulating Equations: The " }{TEXT 338 6 "lhs( )" }{TEXT -1 5 " and "
}{TEXT 339 6 "rhs( )" }{TEXT -1 9 " commands" }}{PARA 0 "" 0 "" {TEXT
340 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 90 "Recall that we can \+
give a name to an entire equation just as we have done for expressions
." }}{PARA 0 "" 0 "" {TEXT -1 40 "On the next line we enter the equati
on " }{XPPEDIT 18 0 "x^3-5*x^2+23=2*x^2+4*x-8 " "6#/,(*$%\"xG\"\"$\"
\"\"*&\"\"&F(*$F&\"\"#F(!\"\"\"#BF(,(*&F,F(*$F&F,F(F(*&\"\"%F(F&F(F(\"
\")F-" }{TEXT -1 31 " and give it the name \"eqn1\" ." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn1:=x
^3-5*x^2+23=2*x^2+4*x-8;" }}}{PARA 0 "" 0 "" {TEXT 341 10 "Example 2:
" }}{PARA 0 "" 0 "" {TEXT -1 79 "We can isolate the left-hand and righ
t-hand sides of the equation by using the " }{TEXT 342 6 "lhs( )" }
{TEXT -1 5 " and " }{TEXT 343 6 "rhs( )" }{TEXT -1 11 " commands. " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lhs(eqn1);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 10 "rhs(eqn1);" }}}{PARA 0 "" 0 "" {TEXT 344
9 "Example 3" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 8 "Use the "
}{TEXT 345 6 "lhs( )" }{TEXT -1 5 " and " }{TEXT 346 6 "rhs( )" }
{TEXT -1 144 " commands to find an equation that is equivalent to the \+
original equation eqn1 but has zero on the right-hand side. Label the \+
new equation eqn2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn2:=
lhs(eqn1)-rhs(eqn1)=0;" }}}}{SECT 1 {PARA 4 "" 0 "solve( )" {TEXT -1
29 "Finding Exact Solutions: The " }{TEXT 327 8 "solve( )" }{TEXT -1
8 " command" }}{PARA 0 "" 0 "" {TEXT -1 18 "We first consider " }
{TEXT 279 10 "polynomial" }{TEXT -1 49 " equations. Algorithms exist f
or calculating the " }{TEXT 284 5 "exact" }{TEXT -1 15 " solutions for
" }{TEXT 285 10 "polynomial" }{TEXT -1 17 " equations up to " }{TEXT
286 8 "degree 4" }{TEXT -1 10 ". Maple's " }{TEXT 297 8 "solve( )" }
{TEXT -1 38 " command implements these algorithms. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 10 "Example 1:" }}{PARA 0
"" 0 "" {TEXT -1 56 "To find the exact solutions to the polynomial equ
ation " }{XPPEDIT 18 0 "3*x^3-4*x^2-43*x+84=0" "6#/,**&\"\"$\"\"\"*$%
\"xGF&F'F'*&\"\"%F'*$F)\"\"#F'!\"\"*&\"#VF'F)F'F.\"#%)F'\"\"!" }{TEXT
-1 9 " use the " }{TEXT 260 9 "solve( ) " }{TEXT -1 10 "command. " }}
{PARA 0 "" 0 "" {TEXT -1 114 "Note that the second argument of the com
mand tells Maple that x is the unknown variable that we are solving fo
r. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(3*x^3-4*x^2-43
*x+84=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 66 "Here Maple has found all \+
three solutions and listed them for you. " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT 280 10 "Example 2:" }}{PARA 0 "" 0 ""
{TEXT -1 172 "Sometimes you will want to select one solution from the \+
list of solutions and use it in another computation. You can do this \+
by first assigning a name (we use the letter N" }{TEXT 313 1 " " }
{TEXT -1 35 "in this case) to the output of the " }{TEXT 261 8 "solve(
)" }{TEXT -1 15 " command. Then " }{TEXT 258 4 "N[1]" }{TEXT -1 34 " \+
is the first number in the list, " }{TEXT 259 4 "N[2]" }{TEXT -1 59 " \+
is the second number and so on. Note the square brackets." }}{PARA 0
"" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "N:=
solve(x^2-5*x+3=0,x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N
[1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 10
"Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 22 "When working with the " }
{TEXT 302 8 "solve( )" }{TEXT -1 75 " command it is often convenient t
o begin by giving a name to the equation. " }}{PARA 0 "" 0 "" {TEXT
-1 77 "Note we use \" := \" to assign the name and just \"=\" for th
e equation itself." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn1:=
7*x^3-11*x^2-27*x-9=0;" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Next we solve \+
the equation using the " }{TEXT 273 8 "solve( )" }{TEXT -1 44 " comman
d assigning the name H to the output." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "H:=solve(eqn1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 104 "F
or practice let's check that each of these values satisfies the equati
on. This is easy to do using the " }{TEXT 272 7 "subs( )" }{TEXT -1 9
" command." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=H[1],eq
n1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=H[2],eqn1);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=H[3],eqn1);" }}}
{PARA 0 "" 0 "" {TEXT 282 10 "Example 4:" }}{PARA 0 "" 0 "" {TEXT -1
114 "Sometimes the \"exact\" solutions are too cumbersome to be of muc
h use. In the next two lines we solve the equation " }{XPPEDIT 18 0 "x
^3-34*x^2+4=0" "6#/,(*$%\"xG\"\"$\"\"\"*&\"#MF(*$F&\"\"#F(!\"\"\"\"%F(
\"\"!" }{TEXT -1 3 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e
qn1:=x^3-34*x^2+4=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "H:=solve(eq
n1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "As you can see, reading
these exact solutions is quite a challenge! Note that the " }{TEXT
361 1 "I" }{TEXT -1 13 " stands for " }{XPPEDIT 18 0 "sqrt(-1)" "6#-%
%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 99 ". When a solution is this complic
ated it is more useful to look at the approximate solutions using " }
{TEXT 360 8 "evalf( )" }{TEXT -1 4 ": . " }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 9 "evalf(H);" }}}{PARA 0 "" 0 "" {TEXT -1 26 "A good alte
rnative to the " }{TEXT 287 8 "solve( )" }{TEXT -1 41 " command in a s
ituation like this is the " }{TEXT 288 9 "fsolve( )" }{TEXT -1 54 " co
mmand which will be discussed in the next section. " }}{PARA 0 "" 0 "
" {TEXT -1 4 "The " }{TEXT 301 8 "solve( )" }{TEXT -1 58 " command can
also be used to find the exact solutions for " }{TEXT 257 14 "non-pol
ynomial" }{TEXT -1 243 " equations. Some simple examples are listed be
low. However if the equations are at all complicated, for example comb
ining exponential, polynomial and trigonometric expressions, then an e
xact solution will typically not be available. Again the " }{TEXT 300
9 "fsolve( )" }{TEXT -1 28 " command is an alternative. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 10 "Example 5:" }}
{PARA 0 "" 0 "" {TEXT -1 23 "Solve the equation: " }{XPPEDIT 18 0 "
5*exp(x/4)=43 " "6#/*&\"\"&\"\"\"-%$expG6#*&%\"xGF&\"\"%!\"\"F&\"#V" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(5*exp(x/4)=43,x);" }}
}{PARA 0 "" 0 "" {TEXT 299 10 "Example 6:" }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Sometimes Maple does not display \+
" }{TEXT 262 3 "all" }{TEXT -1 112 " of the solutions. How would you u
se the result below to write down the entire set of solutions to the e
quation?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(sin(x)=1/2
,x);" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 271 12 "Exercise 4.1" }}{PARA
0 "" 0 "" {TEXT -1 22 "Solve the equation " }{XPPEDIT 18 0 "x^3-11*
x^2+7*x+147 = 0." "6#/,**$%\"xG\"\"$\"\"\"*&\"#6F(*$F&\"\"#F(!\"\"*&\"
\"(F(F&F(F(\"$Z\"F($\"\"!F2" }{TEXT -1 165 " Why does Maple produce o
nly two distinct solutions for this cubic equation? Why is one of the
m written twice? (HINT: Factor the left hand side of the equation.)
" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 4.1" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 314 10 "Answer \+
4.1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(x^3-11*x^2+7*x+
147=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "factor(x^3-11*
x^2+7*x+147);" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The fact that x - 7 is \+
a " }{TEXT 263 16 "repeated factor " }{TEXT -1 99 "results in the cubi
c equation having only two distinct roots, -3 and 7. We say that roo
t 7 has a " }{TEXT 264 17 "multiplicity of 2" }{TEXT -1 21 ", meaning
there are " }{TEXT 265 3 "two" }{TEXT -1 48 " factors of (x - 7) in \+
the factored polynomial." }}}}}{SECT 1 {PARA 4 "" 0 "fsolve( )"
{TEXT -1 35 "Finding Approximate Solutions: The " }{TEXT 328 9 "fsolve
( )" }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }
{TEXT 306 9 "fsolve( )" }{TEXT -1 73 " command can be used to find app
roximate solutions for any equation. For " }{TEXT 308 10 "polynomial"
}{TEXT -1 11 " equations " }{TEXT 307 8 "fsolve()" }{TEXT -1 12 " prod
uces a " }{TEXT 315 13 "complete list" }{TEXT -1 79 " of all of the re
al solutions in one step (see Example 1). For other equations " }
{TEXT 309 9 "fsolve( )" }{TEXT -1 20 " can be used to get " }{TEXT
316 22 "one solution at a time" }{TEXT -1 25 " (see Examples 2 and 3).
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 10 "Exa
mple 1:" }}{PARA 0 "" 0 "" {TEXT -1 9 "Maple's " }{TEXT 304 9 "fsolve
( )" }{TEXT -1 155 " command will compute a numerical approximation fo
r each of the real solutions of a polynomial equation. Approximate all
real solutions for the equation: " }{XPPEDIT 18 0 "x^4-x^3-17*x^2-6*
x+2=0" "6#/,,*$%\"xG\"\"%\"\"\"*$F&\"\"$!\"\"*&\"# " 0 "" {MPLTEXT 1 0 28 "eqn:=x^4-x^3-17*x^2-6*x+
2=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x);" }}}
{PARA 0 "" 0 "" {TEXT -1 114 "The four solutions listed above provide \+
us with a complete list of the solutions to the given polynomial equat
ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 10 "
Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{TEXT 319 3 "all" }
{TEXT -1 33 " real solutions of the equation " }{XPPEDIT 18 0 "x^3+1-
exp(x)=0" "6#/,(*$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&!\"\"\"\"!" }{TEXT
-1 12 " using the " }{TEXT 310 9 "fsolve( )" }{TEXT 311 10 " command.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eqn:=x^3+1-exp(x)=0;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x);" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Maple returns \+
" }{TEXT 320 3 "one" }{TEXT -1 206 " real solution. This time Maple ha
s not given us the whole story. Are there any other solutions? How d
o we find them? A systematic procedure for finding the remaining solut
ions is presented in Example 3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 312 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1
49 "Find the other real solutions for the equation " }{XPPEDIT 18 0
"x^3+1-exp(x)=0" "6#/,(*$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&!\"\"\"\"!" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT 337 103 "The first step in finding the other solutions is to plo
t a graph of the left-hand side of the equation." }}{PARA 0 "" 0 ""
{TEXT 336 11 "Key Concept" }{TEXT -1 18 ": Recall that the " }{TEXT
362 13 "x-intercepts " }{TEXT -1 4 "of " }{XPPEDIT 18 0 "y=x^3+1-exp(
x)" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"F)F)-%$expG6#F'!\"\"" }{TEXT -1 54 " \+
correspond exactly to the solutions of the equation " }{XPPEDIT 18 0
"x^3+1-exp(x)=0" "6#/,(*$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&!\"\"\"\"!" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 36 "plot(x^3+1-exp(x),x=-3..5,y=-5..15);" }}}{PARA
0 "" 0 "" {TEXT -1 16 "The graph shows " }{TEXT 317 4 "four" }{TEXT
-1 90 " x-intercepts. One of these corresponds to the solution we foun
d in Example 2. Which one? " }}{PARA 0 "" 0 "" {TEXT -1 70 "The x=0 so
lution is also easy to spot. How do we find the other three?" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can extend t
he " }{TEXT 321 9 "fsolve( )" }{TEXT -1 139 " command to look for a so
lution in a particular interval. For example to find the negative solu
tion we ask Maple to search on the interval " }{XPPEDIT 18 0 "[ -1 , -
0.2]" "6#7$,$\"\"\"!\"\",$$\"\"#F&F&" }{TEXT -1 104 " since we can see
from the graph that there definitely is one (and only one) solution o
n that interval. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "fsolve(eqn,x=-1..-.2);" }}}{PARA 0 "" 0 ""
{TEXT -1 39 "To find the other two solutions we use " }{TEXT 318 9 "fs
olve( )" }{TEXT -1 75 " again, this time with search interval [1,2] an
d then with interval [4,5]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
39 "fsolve(eqn,x=1..2);\nfsolve(eqn,x=4..5);" }}}{PARA 0 "" 0 ""
{TEXT -1 97 "What happens if you ask Maple to search for a solution on
an interval where no solution exists ? " }}{PARA 0 "" 0 "" {TEXT -1
121 "Let's try it out. From the graph it is clear that there are no x-
intercepts (and therefore no solutions) between 2 and 4." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(eqn,x=2..4);" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Notice that Maple s
imply returns the original input line unchanged when it cannot find a \+
solution on the given interval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 419 "Are there any other solutions? For exam
ple, are there any solutions larger than 5 ? We can check this out by
expanding the interval over which the graph is plotted. On the next l
ine we expand the interval to [-3,50]. No other x-intercepts appear. T
he graph confirms what we should expect by looking at the terms of the
expression, namely the exponential term dominates and causes the grap
h to go down in the long run." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 38 "plot(x^3+1-exp(x),x=-3..50,y=-10..15);" }}}{PARA 0 "" 0 ""
{TEXT -1 88 "Alternatively we can use the fsolve( ) command, now searc
hing over this larger interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 20 "fsolve(eqn,x=5..50);" }}}{PARA 0 "" 0 "" {TEXT -1 45 "As expec
ted no solutions are found by Maple. " }}{PARA 0 "" 0 "" {TEXT -1 51 "
In a similar way we can check for solutions to the " }{TEXT 358 4 "lef
t" }{TEXT -1 59 ". Here we search for solutions over the interval [-50
,-1] ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 22 "fsolve(eqn,x=-50..-1);" }}}{PARA 0 "" 0 "" {TEXT -1
19 "None there either !" }}{PARA 0 "" 0 "" {TEXT -1 75 "We now have a \+
complete list of the four solutions of our original equation " }
{XPPEDIT 18 0 "x^3+1-exp(x)=0" "6#/,(*$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&
!\"\"\"\"!" }{TEXT -1 62 " . They are: -.8251554597 , 0 , 1.54500727
9 and 4.567036837" }}{PARA 0 "" 0 "" {TEXT 333 10 "Example 4:" }}
{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 334 9 "fsolve( )" }{TEXT -1
55 " to find the approximate solutions of the equation : " }
{XPPEDIT 18 0 "x^2/20-10*x=15*cos(x+15)" "6#/,&*&%\"xG\"\"#\"#?!\"\"\"
\"\"*&\"#5F*F&F*F)*&\"#:F*-%$cosG6#,&F&F*F.F*F*" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 250 "Just as in the last exam
ple we will use a graph to help us determine the number and approximat
e location of the solutions. Our task is simplified if we start by con
verting the given equation to an equivalent one that has zero on the r
ight-hand side. " }}{PARA 0 "" 0 "" {TEXT -1 45 "So we will solve the \+
equivalent equation : " }{XPPEDIT 18 0 "x^2/20-10*x-15*cos(x+15)=0"
"6#/,(*&%\"xG\"\"#\"#?!\"\"\"\"\"*&\"#5F*F&F*F)*&\"#:F*-%$cosG6#,&F&F*
F.F*F*F)\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 20 "If we now graph the " }{TEXT 335 14 "left-hand side" }
{TEXT -1 81 " of this equation we once again will find solutions at ea
ch of the x-intercepts. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "
eqn:=x^2/20-10*x-15*cos(x+15)=0;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "plot(lhs(eqn),x=-10..10);" }}}{PARA 0 "" 0 "" {TEXT
-1 74 "From the graph it appears that there is a solution on the inter
val [1,2]. " }}{PARA 0 "" 0 "" {TEXT -1 62 "We now direct Maple to sea
rch for a solution on this interval." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 19 "fsolve(eqn,x=1..2);" }}}{PARA 0 "" 0 "" {TEXT -1 172
"Have we found all of the solutions to this equation? In fact there i
s another solution! To find it start by expanding the interval over wh
ich the graph is drawn. Then use " }{TEXT 359 9 "fsolve( )" }{TEXT -1
60 " to find a numerical approximation for this second solution." }}
{SECT 1 {PARA 4 "" 0 "" {TEXT 326 12 "Exercise 4.2" }}{PARA 0 "" 0 ""
{TEXT -1 39 "Find all the solutions to the equation " }{XPPEDIT 18 0 "
x^5-4*x^3+3*x^2+7*x-1=0" "6#/,,*$%\"xG\"\"&\"\"\"*&\"\"%F(*$F&\"\"$F(!
\"\"*&F,F(*$F&\"\"#F(F(*&\"\"(F(F&F(F(F(F-\"\"!" }{TEXT -1 41 ". Begin
by looking at a relevant graph. " }}{SECT 1 {PARA 5 "" 0 "" {TEXT
-1 21 "Student Workspace 4.2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "
" 0 "" {TEXT -1 10 "Answer 4.2" }}{PARA 0 "" 0 "" {TEXT -1 56 "We begi
n by graphing the left-hand side of the equation." }}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 29 "eqn:=x^5-4*x^3+3*x^2+7*x-1=0;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(lhs(eqn),x=-5..5,y=-5..5);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "This picture indicates that ther
e are solutions near -2 , -1.5, and 0. We next try the unrestricted \+
" }{TEXT 325 6 "fsolve" }{TEXT -1 46 " command to see which solution(s
) Maple finds." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eq
n,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Since this is a polynom
ial equation the fsolve( ) command gives us a complete list of the rea
l solutions." }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 324 12 "Exercise 4.3
" }}{PARA 0 "" 0 "" {TEXT -1 39 "Find all the solutions to the equatio
n " }{XPPEDIT 18 0 "x^2 - 2 = ln(x+5)" "6#/,&*$%\"xG\"\"#\"\"\"F'!\"\"
-%#lnG6#,&F&F(\"\"&F(" }{TEXT -1 20 ". Use the graph of " }{TEXT 322
3 "one" }{TEXT -1 107 " expression to locate the solutions. Check eac
h solution by substituting it back in the original equation." }}{SECT
1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 4.3" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 10
"Answer 4.3" }}{PARA 0 "" 0 "" {TEXT -1 75 "First we put the equation \+
in \"standard form\" , i.e. with zero on one side. " }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 21 "eqn:=x^2-2-ln(x+5)=0;" }}}{PARA 0 "" 0 ""
{TEXT -1 46 "Now we can graph the right-hand side equation." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(lhs(eqn),x=-10..10);" }}}
{PARA 0 "" 0 "" {TEXT -1 73 "There appear to be two solutions. One ne
ar -2 and the other near 2. Use " }{TEXT 323 6 "fsolve" }{TEXT -1 83 "
with a restricted domain to find the two solutions you've located mor
e precisely. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "soln1:=fsol
ve(eqn,x=-5..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "soln2:=
fsolve(eqn,x=1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 53 "Check by subsitut
ing back into the original equation." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "evalf(subs(x=soln1,eqn));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "evalf(subs(x=soln2,eqn));" }}}{PARA 0 "" 0 ""
{TEXT -1 178 "Notice that the original equation is only \"approximatel
y\" satisfied by each of our solutions. The slight discrepancy is a r
esult of round-off error in the approximate solutions." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Check that there are
no additional solutions." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 357 12 "E
xercise 4.4" }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " }
{XPPEDIT 18 0 "y=10-x^2" "6#/%\"yG,&\"#5\"\"\"*$%\"xG\"\"#!\"\"" }
{TEXT -1 7 " and " }{XPPEDIT 18 0 "y=4*sin(2*x)+5" "6#/%\"yG,&*&\"\"
%\"\"\"-%$sinG6#*&\"\"#F(%\"xGF(F(F(\"\"&F(" }{TEXT -1 41 " intersect \+
twice on the interval [-5,5]. " }}{PARA 0 "" 0 "" {TEXT -1 90 "a) Grap
h the two equations together and estimate the intersection points usin
g the mouse. " }}{PARA 0 "" 0 "" {TEXT -1 94 "b) Write an equation tha
t can be solved to find the x-coordinates of the intersection points. \+
" }}{PARA 0 "" 0 "" {TEXT -1 40 "c) Use fsolve( ) to solve this equati
on." }}{PARA 0 "" 0 "" {TEXT -1 89 "d) Use the results from part c) to
estimate the y-coordinates of the intersection points." }}{PARA 0 ""
0 "" {TEXT -1 180 "e) It appears that the curves may intersect at a th
ird point near (1,9). Use fsolve( ) and/or a relevant graph to demonst
rate that there is no intersection point at that location. " }}{SECT
1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 4.4" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 10
"Answer 4.4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "y1:=10-x^2;"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "y2:=4*sin(2*x)+5;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Here is a plot of the two equation
s. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot([y1,y2],x=-5..5
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Intersection points are loc
ated approximately at: (-1.8, 6.6) and ( 2.75, 2) . " }}{PARA 0 "" 0 "
" {TEXT -1 34 "b) The equation to solve is y1=y2." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "eqn:= y1=y2;" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 49 "c) We now find the two solutions using fsolve( ) " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x_soln1:=fsolve(y1=y2,x=-4..
0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x_soln2:=fsolve(y1=y
2,x=0..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "d) We can use the s
ubs( ) command to find the corresponding y-coordinates." }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "y_soln1:=subs(x=x_soln1,y1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "y_soln2:=subs(x=x_soln2,y1);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "So the points of intersection
are : (-1.800,6.763) and (2.773,2.311)" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 56 "e) Here is a closer look at what is happening near x=1. \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot([y1,y2],x=.5..1.5)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Alternatively we can use fso
lve( ) to confirm that there is no solution near x=1." }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(y1=y2,x=.5..1.5);" }}}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "
Solving Literal Equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
restart:" }}}{PARA 0 "" 0 "" {TEXT -1 70 "Often Maple can solve litera
l equations for any one of the variables. " }}{PARA 0 "" 0 "" {TEXT
-1 61 "Suppose we want to solve for the variable g in the equation: "
}{XPPEDIT 18 0 "4-v=2*T-k*g" "6#/,&\"\"%\"\"\"%\"vG!\"\",&*&\"\"#F&%\"
TGF&F&*&%\"kGF&%\"gGF&F(" }{TEXT -1 6 " . " }}{PARA 0 "" 0 ""
{TEXT -1 36 "The solve command works well here. " }}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 21 "solve(4-v=2*T-k*g,g);" }}}{PARA 0 "" 0 ""
{TEXT -1 57 "Here is a little nicer way of displaying the same result:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g=solve(4-v=2*T-k*g,g);
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 291 12 "Exercise 4.4" }}{PARA 0 ""
0 "" {TEXT -1 61 "Edit the last command to solve for each of the other
letters " }{TEXT 289 14 "T, k and v. " }}{SECT 1 {PARA 20 "" 0 ""
{TEXT 292 21 "Student Workspace 4.4" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 20 "" 0 "" {TEXT 293 10 "Answer 4.4" }}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 23 "T=solve(4-v=2*T-k*g,T);" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 23 "k=solve(4-v=2*T-k*g,k);" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 23 "v=solve(4-v=2*T-k*g,v);" }}}}}{SECT 1 {PARA
4 "" 0 "" {TEXT 294 12 "Exercise 4.5" }}{PARA 0 "" 0 "" {TEXT -1 19 "S
olve the equation " }{XPPEDIT 18 0 "x^2+y^2=9" "6#/,&*$%\"xG\"\"#\"\"
\"*$%\"yGF'F(\"\"*" }{TEXT -1 5 " for " }{TEXT 290 1 "y" }{TEXT -1
104 ". Assign the set of solutions to a variable named S. How are the
two solutions S[1] and S[2] related ? " }}{SECT 1 {PARA 20 "" 0 ""
{TEXT 295 21 "Student Workspace 4.5" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 20 "" 0 "" {TEXT 296 10 "Answer 4.5" }}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 23 "S:=solve(x^2+y^2=25,y);" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 5 "S[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 5 "S[2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The solution S[1] i
s the negative of solution S[2]." }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Solving a Linear System of Equa
tions using the " }{TEXT 329 8 "solve( )" }{TEXT -1 9 " command " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 352 52 "Please e
xecute the next two lines before proceeding:" }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
12 "with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 4 "The " }{TEXT 266 8 "solve( )" }{TEXT -1 69 " command can a
lso be used to solve a system of m linear equations in " }{TEXT 268 1
"n" }{TEXT -1 28 " variables. We call these " }{TEXT 269 5 "m by " }
{TEXT 267 1 "n" }{TEXT 270 15 " linear systems" }{TEXT -1 11 " for sho
rt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 351 10 "E
xample 1:" }}{PARA 0 "" 0 "" {TEXT -1 28 "Solve the 2 by 2 system: \+
" }{XPPEDIT 18 0 "3*x+2*y=3" "6#/,&*&\"\"$\"\"\"%\"xGF'F'*&\"\"#F'%\"y
GF'F'F&" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "x-y=-4" "6#/,&%\"xG\"\"
\"%\"yG!\"\",$\"\"%F(" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 26 "solve(\{3*x+2*y=3,x-y=-4\});" }}}{PARA 0 "" 0 ""
{TEXT -1 247 "A graph of the two underlying functions shows the soluti
on corresponds to the point of intersection at (-1,3). But we first n
eed to find the explicit form for each of the linear functions before \+
we can graph them. So we solve each equation for y." }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 23 "y1:=solve(3*x+2*y=3,y);" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 20 "y2:=solve(x-y=-4,y);" }}}{PARA 0 "" 0 ""
{TEXT -1 217 "Now we construct a picture made up of two parts: \"part1
\" contains the graphs the two equations and \"part2\" plots the solut
ion point that we found. This point should be the intersection point o
f the two lines. Is it ? " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32
"part1:=plot([y1,y2],x=-5..5): " }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 59 "part2:=plot([[-1,3]],style=point,color=blue,symbol=ci
rcle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display([part1,pa
rt2]);" }}}{PARA 0 "" 0 "" {TEXT 353 10 "Example 2:" }}{PARA 0 "" 0 "
" {TEXT -1 40 "Here is an example of the solution of a " }{TEXT 354 6
"3 by 3" }{TEXT -1 35 " system with variables x, y, and z." }}{PARA 0
"" 0 "" {TEXT -1 27 "Solve the 3 by 3 system: " }{XPPEDIT 18 0 "\{x+
y+z=1, 3*x+y=3, x-2*y-z=0\}" "6#<%/,(%\"xG\"\"\"%\"yGF'%\"zGF'F'/,&*&
\"\"$F'F&F'F'F(F'F-/,(F&F'*&\"\"#F'F(F'!\"\"F)F2\"\"!" }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(\{x
+y+z=1, 3*x+y=3, x-2*y-z=0\});" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 274
12 "Exercise 4.6" }}{PARA 0 "" 0 "" {TEXT -1 32 "Find the solution to \+
the system:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "4*x+3*y=12 and 5*x-7*y=3
5" "6#3/,&*&\"\"%\"\"\"%\"xGF(F(*&\"\"$F(%\"yGF(F(\"#7/,&*&\"\"&F(F)F(
F(*&\"\"(F(F,F(!\"\"\"#N" }}{PARA 0 "" 0 "" {TEXT -1 72 "Check by subs
itituting the solution pair in both equations in the system" }}{SECT
1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 4.6" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1
{PARA 5 "" 0 "" {TEXT -1 10 "Answer 4.6" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 31 "eqns:=\{4*x+3*y=12, 5*x-7*y=35\};" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 17 "ans:=solve(eqns);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 29 "subs(x=189/43,y=-80/43,eqns);" }}}}}{SECT 1
{PARA 4 "" 0 "" {TEXT -1 62 "Linear Systems with an Infinite Number of
Solutions (Optional)" }}{PARA 0 "" 0 "" {TEXT -1 23 "When a system ha
s more " }{TEXT 355 9 "variables" }{TEXT -1 6 " than " }{TEXT 356 9 "e
quations" }{TEXT -1 59 " we often get not one, but an infinite number \+
of solutions." }}{PARA 0 "" 0 "" {TEXT -1 19 "Here is an example." }}
{PARA 0 "" 0 "" {TEXT 349 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1
19 "Solve the system : " }{XPPEDIT 18 0 "\{ x+y+z=1 , 3*x+y=3 \}" "6#<
$/,(%\"xG\"\"\"%\"yGF'%\"zGF'F'/,&*&\"\"$F'F&F'F'F(F'F-" }{TEXT -1 1 "
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solns:=solve(\{x+y+z=1,
3*x+y=3\});" }}}{PARA 0 "" 0 "" {TEXT -1 180 "Notice this time we do \+
not get a single set of numerical values for x, y and z. Instead Maple
tells us how the values of x, y and z must be related to construct a \+
typical solution." }}{PARA 0 "" 0 "" {TEXT -1 29 "In particular the ex
pression " }{XPPEDIT 18 0 "x=x" "6#/%\"xGF$" }{TEXT 347 1 " " }{TEXT
-1 35 "in the output above indicates that " }{XPPEDIT 18 0 "x" "6#%\"x
G" }{TEXT -1 8 " can be " }{TEXT 348 3 "any" }{TEXT -1 76 " number. We
refer to it as the \"free\" variable in the solution. To find any " }
{TEXT 350 10 "particular" }{TEXT -1 154 " solution (among the infinte \+
number possible) pick any value for x and use this to calculate the co
rresponding values for y and z. For example let x = 4. " }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=4,solns); " }}}{PARA 0 ""
0 "" {TEXT -1 146 "So one solution is : x=4 , y= -9 and z=6. Take a m
inute and check by hand that these three numbers do in fact satisfy ou
r original equations : " }{XPPEDIT 18 0 " x+y+z=1" "6#/,(%\"xG\"\"\"
%\"yGF&%\"zGF&F&" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "3*x+y=3" "6#/,&
*&\"\"$\"\"\"%\"xGF'F'%\"yGF'F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 ""
{TEXT -1 66 "Now let's look at the solution that is generated when we \+
take x=2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(x=2,solns)
;" }}}{PARA 0 "" 0 "" {TEXT -1 83 "So two of the infinitely many solut
ions are: (x,y,z) = (4, -9, 6) and (2, -3, 2). " }}{SECT 1 {PARA 4 "
" 0 "" {TEXT 330 12 "Exercise 4.7" }}{PARA 0 "" 0 "" {TEXT -1 19 "Solv
e the system : " }{XPPEDIT 18 0 "\{ x+2*y+z=2 , 3*x+y=1 \}" "6#<$/,(%
\"xG\"\"\"*&\"\"#F'%\"yGF'F'%\"zGF'F)/,&*&\"\"$F'F&F'F'F*F'F'" }{TEXT
-1 46 " and find at least three particular solutions." }}{SECT 1
{PARA 20 "" 0 "" {TEXT 331 21 "Student Workspace 4.7" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 332 10 "Answer 4.7" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqns:=\{ x+2*y+z=2 , 3*x+y=1
\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "soln:=solve(eqns);
" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Here are the solutions for x = 1, 2,
3, and 4 :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=1,soln
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=2,soln);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=3,soln);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=4,soln);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2
33 1 1 }