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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 276 66 "Laboratoire 8 - La m
\351thode d'\351limination de Gauss, \351tape par \351tape" }}}{EXCHG 
{PARA 258 "" 0 "" {TEXT -1 52 "Par Claude St-Hilaire, claude.sthilaire
@videotron.ca" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:wi
th(linalg): " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 282 51 "Principales com
mandes utilis\351es dans ce laboratoire" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 36 "swaprow, mulrow, addrow, genmatrix, " }{TEXT 283 5 "pivot
" }{TEXT 284 2 ", " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Note
 : S\351lectionner un mot et utiliser l'aide pour plus d'information" 
}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 55 "\311chelonner une matrice av
ec les op\351rations \351l\351mentaires" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 205 "Un syst\350me de m \351quations \340 n variables peut s'
\351crire sous la forme Ax = b o\371 A est la matrice des coefficients
 des variables, x le vecteur-colonne des variables et b est le vecteur
-colonne des constantes." }}{PARA 0 "" 0 "" {TEXT -1 57 "La m\351thode
 d'\351limination de Gauss consiste \340 transformer " }{TEXT 268 28 "
la matrice augment\351e [A | b]" }{TEXT -1 121 ", en appliquant les op
\351rations \351l\351mentaires sur les lignes, en une matrice \351chel
onn\351e [H | c] ayant les m\352mes solutions. " }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 260 10 "gausselim(" }{TEXT 270 17 "matrice augment\351e" }
{TEXT 269 1 ")" }{TEXT -1 10 " donne la " }{TEXT 262 28 "matrice \351c
helonn\351e [H | c] ." }{TEXT -1 32 " Cependant , on peut construire \+
" }{TEXT 261 15 "\351tape par \351tape" }{TEXT -1 39 " cette matrice \+
\351chelonn\351e en utilisant " }{TEXT 272 44 "les 3 op\351rations \+
\351l\351mentaires sur les lignes" }{TEXT -1 12 " suivantes :" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "1) " }{TEXT 257 14 "swaprow(M,i,j)
" }{TEXT -1 86 " est \351quivalent \340  Li <->Lj : interchange la lig
ne i avec la ligne j dans la matrice M" }}{PARA 0 "" 0 "" {TEXT -1 3 "
2) " }{TEXT 258 13 "mulrow(M,i,k)" }{TEXT -1 75 " est \351quivalent \+
\340  Li ->k Li  : multiplie par k, la ligne i de la matrice M" }}
{PARA 0 "" 0 "" {TEXT -1 3 "3) " }{TEXT 259 15 "addrow(M,i,j,k)" }
{TEXT -1 165 " est \351quivalent  \340  Lj -> Lj + k Li , avec la matr
ice M, c'est-\340-dire multiplier la ligne i par k et additionner term
e \340 terme \340 la ligne j pour remplacer la ligne j" }}{PARA 0 "" 
0 "" {TEXT -1 29 "Illustrons ces 3 op\351rations :" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "A:=matrix(3,3,[-20,20,50,5,-1,3,-4,9,-8]);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Permutons la ligne 1 avec la \+
ligne 2: L1<->L2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "B:=swap
row(A,1,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Multiplions la tro
isi\350me ligne de la matrice B par 100: L1->(10)L1" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "B:=mulrow(B,3,100);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 96 "Multiplions la ligne 1 de la derni\350re matrice B
 par 4 et additionnons \340 la ligne 2: L2->L2 + 4L1" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "addrow(B,1,2,4);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 275 7 "Exemple" }{TEXT -1 1 " " }{TEXT 277 3 "1 :" }{TEXT -1 
137 "Trouver la matrice \351chelonn\351e du syst\350me d'\351quations \+
lin\351aires suivant en appliquant la m\351thode d'\351limination de G
auss, \351tape par \351tape. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 68 "eq1:=5*y - 7*z = 8;eq2:=-2*x + 8*y +z = 12;eq3:=3*x + 4*y + 5*z \+
= 6;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "La matrice augment\351e M
 est : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "M:=genmatrix([eq
1,eq2,eq3],[x,y,z],flag);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Perm
utons la ligne 1 avec la ligne 2: L1<->L2" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "M1:=swaprow(M,1,2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 94 "Multiplions la ligne 1 de la matrice M1 par 3/2 et additi
onnons \340 la ligne 3: L3->L3 + (3/2)L1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "M2:=addrow(M1,1,3,3/2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 121 "Remarque : Si, \340 l'aide du pivot \"a\" sur la ligne  \+
Li, on veut \351liminer \"b\" sur la ligne Lj, on fait Lj ->Lj + (-b/a
) Li " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Avec le pivot 5 \340 la
 position (2,2), \351liminons 16 \340 la position (3,2) en multilpiant
 la ligne 2 par -16/5 et en additionnant \340 la ligne 3: L3->L3 +(-16
/5)L2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "M3:=addrow(M2,2,3,
-16/5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "M3 est la matrice M \+
\351chelonn\351e: Si on veut la solution du syst\350me d'\351quations,
 utilisons backsub sur la matrice \351chelonn\351e M3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "backsub(M3);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 42 "donc x = 18/289, y = 440/289, z  = -16/289" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Rappel: la matrice \351chelonn\351
e peut \352tre obtenue avec gausselim :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "C:=gausselim(M);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 263 75 "Matrice \+
\351chelonn\351e-r\351duite : Gauss-Jordan, avec les op\351rations \+
\351l\351mentaires" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 293 "Rappel : \+
\300 l'aide des op\351rations \351l\351mentaires sur les lignes d'une \+
matrice augment\351e, la m\351thode de Gauss consiste \340 mettre des \+
z\351ros sous les pivots pour obtenir une matrice \351chelonn\351e. Si
 en plus, on met des z\351ros au dessus des pivots et qu'on rend les p
ivots unitaires(=1) alors on obtient " }{TEXT 274 2 "sa" }{TEXT -1 1 "
 " }{TEXT 273 26 "matrice \351chelonn\351e-r\351duite" }{TEXT -1 78 ".
 C'est la m\351thode de Gauss-Jordan. Voir le laboratoire 7 m\351thode
 Gauss-Jordan" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 278 7 "
Exemple" }{TEXT -1 1 " " }{TEXT 279 3 "2 :" }{TEXT -1 119 " \300 parti
r de C ci-haut, compl\351tons l'\351chelonnage de la matrice augment
\351e M, pour obtenir sa matrice \351chelonn\351e-r\351duite " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 " \300 l'aide du pivot 5 \340 la po
sition (2,2) \351liminons 8 \340 la position (1,2)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "addrow(C,2,1,-8/5);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 76 " \300 l'aide du pivot 289/10 la position (3,3) \351l
iminons -7 \340 la position (2,3)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "addrow(%,3,2,7*10/289);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "\300  l'aide du p
ivot 289/10 \340 la position (3,3) \351liminons 61/5 \340 la position \+
(1,3)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "addrow(%,3,1,(-61/
5)/(289/10));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Rendons les pivo
ts \351gaux \340 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "mulr
ow(%,1,-1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "mulrow(%,2
,1/5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "mulrow(%,3,10/289
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "C'est la matrice \351chelon
n\351e-r\351duite de M. " }}{PARA 0 "" 0 "" {TEXT -1 75 "La solution d
u syst\350me d'\351quations : x = 18/289, y = 440/289 et z = -16/289" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "rref (reduce row echelon form) \+
donne la matrice \351chelonn\351e-r\351duite" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "rref(M);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "La derni\350re colonne \+
indique la solution du syst\350me d'\351quations associ\351 \340 la ma
trice augment\351e M. C'est la solution que donne gausselim(M) suivi d
e backsub ou linsolve(A,b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 11 "Ex
emple 3 :" }{TEXT -1 70 " On peut mettre des z\351ros sur une colonne \+
d'une matrice A \340 l'aide du " }{TEXT 271 5 "pivot" }{TEXT -1 1 " " 
}{TEXT 265 25 "A(i,j) en une seule \351tape" }{TEXT -1 7 " avec  " }
{TEXT 264 12 "pivot(A,i,j)" }}{PARA 0 "" 0 "" {TEXT -1 57 "\311chelonn
ons la matrice B suivante avec la commande pivot:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 52 "B:=matrix(3,5,[3,3,6,1,12,-5,8,3,4,-11,-2,7,
5,9,7]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B:=pivot(B,1,1)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B:=pivot(B,2,2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Il n'y a pas de pivot (3,3), allon
s \340 la position (3,4)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"B:=pivot(B,3,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Il ne reste \+
qu'a mettre les pivots \351gaux \340 1 pour obtenir la matrice \351che
lonn\351e r\351duite" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "B:=
mulrow(B,1,1/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B:=mulr
ow(B,2,1/13);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B:=mulrow(
B,3,39/224);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "La matrice de d
\351part est \351chelonn\351e-r\351duite. On n'a pas de pivot sur la t
roisi\350me colonne o\371 on a une variable libre : Si on veut la solu
tion du syst\350me d'\351quations repr\351sent\351 par la matrice augm
ent\351e B, utilisons backsub " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 11 "backsub(B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 266 47 "Syst\350me d'
\351quations lin\351aires avec param\350tres :" }{TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 281 11 "Exemple 4 :" }{TEXT -1 109 " R
\351soudre le syst\350me d'\351quations lin\351aires par la m\351thode
 de Gauss, \351tape par \351tape et avec la commande pivot" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "37*x+19*y+5*z = 4;" "6#/,(*&\"#P
\"\"\"%\"xGF'F'*&\"#>F'%\"yGF'F'*&\"\"&F'%\"zGF'F'\"\"%" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "13*x-12*y+4*z = b;" 
"6#/,(*&\"#8\"\"\"%\"xGF'F'*&\"#7F'%\"yGF'!\"\"*&\"\"%F'%\"zGF'F'%\"bG
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "7*x+3*y+a*z = 12;" "
6#/,(*&\"\"(\"\"\"%\"xGF'F'*&\"\"$F'%\"yGF'F'*&%\"aGF'%\"zGF'F'\"#7" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "eq1:=37*x+19*y+5*z = 4;eq2:=
13*x-12*y+4*z = b;eq3:=7*x+3*y+a*z = 12;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "M:=genmatrix([eq2,eq3,eq1],[x,y,z],flag);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "M1:=pivot(M,1,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M2:=pivot(M1,2,2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "1) Le syst\350me a une solution un
ique si  " }{XPPEDIT 18 0 "703/123-691/123*a;" "6#,&*&\"$.(\"\"\"\"$B
\"!\"\"F&*(\"$\"pF&F'F(%\"aGF&F(" }{TEXT -1 35 " <> 0, c'est-\340-dire
 si a <> 703/691" }}{PARA 0 "" 0 "" {TEXT -1 6 "2) Si " }{XPPEDIT 18 
0 "703/123-691/123*a" "6#,&*&\"$.(\"\"\"\"$B\"!\"\"F&*(\"$\"pF&F'F(%\"
aGF&F(" }{TEXT -1 8 " = 0 et " }{XPPEDIT 18 0 "22/123*b-2600/41;" "6#,
&*(\"#A\"\"\"\"$B\"!\"\"%\"bGF&F&*&\"%+EF&\"#TF(F(" }{TEXT -1 65 " = 0
 alors on a une infinit\351 de solutions (3 variables, 2 pivots)" }}
{PARA 0 "" 0 "" {TEXT -1 7 "3)  Si " }{XPPEDIT 18 0 "703/123-691/123*a
" "6#,&*&\"$.(\"\"\"\"$B\"!\"\"F&*(\"$\"pF&F'F(%\"aGF&F(" }{TEXT -1 8 
" = 0 et " }{XPPEDIT 18 0 "22/123*b-2600/41;" "6#,&*(\"#A\"\"\"\"$B\"!
\"\"%\"bGF&F&*&\"%+EF&\"#TF(F(" }{TEXT -1 87 " <>  0 alors le syst\350
me est incompatible.\nTrouvons les valeurs de a et b qui r\351solvent \+
" }{XPPEDIT 18 0 "703/123-691/123*a" "6#,&*&\"$.(\"\"\"\"$B\"!\"\"F&*(
\"$\"pF&F'F(%\"aGF&F(" }{TEXT -1 9 " = 0 et  " }{XPPEDIT 18 0 "22/123*
b-2600/41" "6#,&*(\"#A\"\"\"\"$B\"!\"\"%\"bGF&F&*&\"%+EF&\"#TF(F(" }
{TEXT -1 4 " = 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a=solve
(703/123-691/123*a=0,a);b=solve(22/123*b-2600/41=0,b);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 184 "Conclusion : on a 1) une solution unique
 si a <> 703/691, 2) une infinit\351 de solutions si a = 703/691 et b \+
= 3900/11 et enfin c) un syst\350me incompatible si a = 703/691 et b <
> 3900/11." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 80 "Trouvons l'infinit\351 de solutions du cas 2), lo
rsque a = 703/691 et b = 3900/11 ?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "subs(\{a=703/691, b=3900/11\},\{eq1,eq2,eq3\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(%,\{x,y,z\});" }}}
{PARA 0 "" 0 "" {TEXT -1 64 "On a bien une infinit\351 de solutions ca
r on a une variable libre." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 267 9 "Exercices" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 291 5 "No 1)" }{TEXT -1 91 " Trouver une matrice A \351c
helonn\351e, \351tape par \351tape, sans utiliser la commande pivot(A,
i,j)," }}{PARA 0 "" 0 "" {TEXT -1 8 " si A = " }{XPPEDIT 18 0 "matrix(
[[2, -5, 7, 0], [9, -4, 11, 6], [5, 7, 1, 0]])" "6#-%'matrixG6#7%7&\"
\"#,$\"\"&!\"\"\"\"(\"\"!7&\"\"*,$\"\"%F+\"#6\"\"'7&F*F,\"\"\"F-" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 285 32 
" Espace de travail de l'\351tudiant" }}{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 "" 
{TEXT 292 5 "No 2)" }{TEXT -1 87 " Trouver la matrice A \351chelonn
\351e-r\351duite du No 1) en utilisant la commande pivot(A,i,j)" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT 286 32 " Espace de travail de l'\351tudi
ant" }}{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 "" {TEXT 293 5 "No 3)" }{TEXT -1 83 " R\351s
oudre le syst\350me d'\351quations lin\351aires par la m\351thode de G
auss, \351tape par \351tape" }}{PARA 0 "" 0 "" {TEXT -1 77 " a) en tro
uvant la matrice \351chelonn\351e sans utiliser la commande pivot(A,i,
j) " }}{PARA 0 "" 0 "" {TEXT -1 48 " b) et en utilisant backsub\n -3x \+
- 4y + 5z  = -5" }}{PARA 0 "" 0 "" {TEXT -1 84 "   x - y - 2z  = -1
\011\011\011\n   -2x + y + z  = -2\011\011\011           \n  -3x + 5y \+
+ 4z  = 3\011\011\011   " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 287 32 " Es
pace de travail de l'\351tudiant" }}{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 "" {TEXT 
294 5 "No 4)" }{TEXT -1 194 " a) R\351soudre le syst\350me d'\351quati
ons lin\351aires par la m\351thode de Gauss, en trouvant la matrice \+
\351chelonn\351e-r\351duite avec la commande pivot(A,i,j)\nb) V\351rif
ier votre r\351ponse avec gausselim et backsub" }}{PARA 0 "" 0 "" 
{TEXT -1 75 "2x1 - 6x2 + 3x3 - 2x4 = -1\n -x1 + 3x2 - 2x3 = 4\n 3x1 - \+
9x2 + 5x3 - 2x4 = -5" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 288 31 "Espace \+
de travail de l'\351tudiant" }}{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 "" {TEXT 295 5 "No \+
5)" }{TEXT -1 94 " R\351soudre le syst\350me d'\351quations homog\350n
es par la m\351thode de Gauss-Jordan, avec pivot(A,i,j). " }}{PARA 0 "
" 0 "" {TEXT -1 48 "V\351rifier votre r\351ponse avec gaussjord et bac
ksub" }}{PARA 0 "" 0 "" {TEXT -1 52 "10x - 3y - 2z = 0\n-7x + 4y + 9z \+
= 0\n11x - 5y - 9z =0" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 289 32 " Espac
e de travail de l'\351tudiant" }}{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 "" {TEXT 296 5 "
No 6)" }{TEXT -1 263 " Par la m\351thode de Gauss, \351tape par \351ta
pe et/ou avec pivot, trouver pour quelles valeurs de a et b le syst
\350me d'\351quations suivant a-t-il \na) une solution unique ?\nb) pa
s de solution ?\nc) une infinit\351 de solutions ?\n2x - 3y + az = -5
\n-2x + 7y - 2z = b\nx  + 3z = -5" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
290 32 " Espace de travail de l'\351tudiant" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "2 0 0" 0 }
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