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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 56 "Systems of linear equat
ions and Gauss elimination method" }}{PARA 0 "" 0 "" {TEXT -1 108 "In \+
this worksheet you will see some example of how to write and solve lin
ear systems with the Maple package." }}{PARA 0 "" 0 "" {TEXT 257 9 "Ex
ample 1" }{TEXT -1 53 ". We solve the following system of linear equat
ions: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solve(\{3*x+y-z=1
,2*x+5*y+2*z=3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "L
et us try another example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "solve(\{x+y=1,2*x+2*y=3\},\{x,y\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 375 "It looks like Maple does not want to solve it. Does that
 mean anything? It is a simple system: we can see that the two equatio
ns have proportional coefficients, however, the right hand side of the
 equations are not proportional with the same constant, so the system \+
is inconsistent. Let's see what we get if we use the elimnation mehod.
 We will use matrix notation this time." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "with(LinearAlgebra);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "A:=<<1,2>|<1,2>>;# the coefficient matrix" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "b:=<<1,3>>;# the right hand side of
 the equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ReducedRo
wEchelonForm(<A|b>);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Now, if \+
we look at the reduced row echaon form of the augmented matrix (<A|b>)
 we see that the second row corresponds to an equation of the form " }
{XPPEDIT 18 0 "0*x+0*y = 1;" "6#/,&*&\"\"!\"\"\"%\"xGF'F'*&F&F'%\"yGF'
F'F'" }{TEXT -1 52 " which is impossible, so the system is inconsisten
t." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 9 "Example 2" }{TEXT -1 179 ".
 Here is another matrix representing the coefficients of a linear syst
em. Write the system corresponding to it and solve the system correspo
nding to the augmented matrix <A1|b1>." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "A1:=<<3,1>|<1,5>|<-1,1>>;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "b1:=<1,3>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "ReducedRowEchelonForm(<A1|b1>);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 259 9 "Example 3" }{TEXT -1 143 ". This example is showing anoth
er way of definig matrices in Maple. However, this format is not appro
priate for defininig the augmented matrix." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "A2:=Matrix([[2,1,0,3],[-1,0,2,4],[4,-2,7,0]]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "b2:=Matrix([[1],[2],[1]]);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ReducedRowEchelonForm(A2)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Same matrix as above defined
 as in Examples 1 and 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 
"A3:=<<2,-1,4>|<1,0,-2>|<0,2,7>|<3,4,0>>;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "b3:=<<1,2,1>>;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "ReducedRowEchelonForm(<A3|b3>);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 260 9 "Example 4" }{TEXT -1 92 ". Let us look at an exerc
ise 8(d) section 1.2 (Anton&Rorres). We define te matrix as before:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "A4:=<<0,1,3,-2,1>|<10,4,2,
-8,-6>|<-4,-1,1,2,3>|<1,1,2,-2,0>>;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "b4:=<1,2,5,-4,1>;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "ReducedRowEchelonForm(<A4|b4>);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 15 "If we want the " }{TEXT 261 8 "solution" }{TEXT 
-1 53 " to the above system we need to call a Maple package:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "linsolve(A4,b4);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "30" 0 }{VIEWOPTS 1 1 0 1 1 
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