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1 }{CSTYLE "" -1 342 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 297 46 "Laboratoire 6 - Methode d'\351limination de Gauss" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 52 "P ar Claude St-Hilaire, claude.sthilaire@videotron.ca" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 306 51 "Principales commandes utilis\351es dans ce lab oratoire" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "matrix de la biblioth \350que linalg, gausselim, backsub, delrows, genmatrix, linsolve, gene qns, augment, delcols, subs avec op, solve, allvalues. " }}{PARA 0 "" 0 "" {TEXT -1 69 "Note : S\351lectionner un mot et utiliser l'aide pou r plus d'information" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 295 7 "Rappels " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "Un syst\350me de m \351quatio ns \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 va riables et b est le vecteur-colonne des constantes." }}{PARA 0 "" 0 " " {TEXT -1 57 "La m\351thode d'\351limination de Gauss consiste \340 t ransformer " }{TEXT 296 28 "la matrice augment\351e [A | b]" }{TEXT -1 123 ", en appliquant les 3 op\351rations \351l\351mentaires sur les lignes, en une matrice \351chelonn\351e [H | c] ayant les m\352mes so lutions. " }}{PARA 0 "" 0 "" {TEXT -1 115 "Le premier nombre non nul d e chaque ligne de la matrice H est appel\351 pivot. Sous un pivot, l' on n'a que des z\351ros." }}{PARA 0 "" 0 "" {TEXT -1 110 "Voir le labo ratoire 7 Gauss\311tapes pour appliquer ces op\351rations \351l\351men taires, \351tape par \351tape, comme \340 la main" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 98 "R\351soudre le syst\350me d'\351quations lin\351ai res suivant \340 l'aide de la m\351thode d'\351limination de Gauss : \+ " }}{PARA 0 "" 0 "" {TEXT -1 52 "Avec Maple : gausselim(matrice augmen t\351e) et backsub" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 273 33 "Cas o \371 il y a une solution unique" }}{EXCHG {PARA 0 "" 0 "" {TEXT 298 11 "Exemple 1 :" }{TEXT -1 11 " R\351soudre :" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "2*x1+3*x2+12*x3-7*x4 = 31;" "6#/,**&\"\"#\"\"\"%#x1G F'F'*&\"\"$F'%#x2GF'F'*&\"#7F'%#x3GF'F'*&\"\"(F'%#x4GF'!\"\"\"#J" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "5*x1+7*x2-4*x3+19*x4 = 3 6;" "6#/,**&\"\"&\"\"\"%#x1GF'F'*&\"\"(F'%#x2GF'F'*&\"\"%F'%#x3GF'!\" \"*&\"#>F'%#x4GF'F'\"#O" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "-8*x1+7*x2-12*x3-11*x4 = -44;" "6#/,**&\"\")\"\"\"%#x1GF'!\"\"*&\" \"(F'%#x2GF'F'*&\"#7F'%#x3GF'F)*&\"#6F'%#x4GF'F),$\"#WF)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "3*x1-2*x2+x3-7*x4 = 15;" "6#/,**& \"\"$\"\"\"%#x1GF'F'*&\"\"#F'%#x2GF'!\"\"%#x3GF'*&\"\"(F'%#x4GF'F,\"#: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "2*x1+10*x2+6*x3+148* x4 = 23;" "6#/,**&\"\"#\"\"\"%#x1GF'F'*&\"#5F'%#x2GF'F'*&\"\"'F'%#x3GF 'F'*&\"$[\"F'%#x4GF'F'\"#B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "-26*x1+80*x2-63*x3+93*x4 = -70;" "6#/,**&\"#E\"\"\"%#x1GF'!\"\"* &\"#!)F'%#x2GF'F'*&\"#jF'%#x3GF'F)*&\"#$*F'%#x4GF'F',$\"#qF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 29 "La matrice aug ment\351e M est : " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "M:=ma trix(6,5,[2,3,12,-7,31,5,7,-4,19,36,-8,7,-12,-11,-44,3,-2,1,-7,15,2,10 ,6,148,23,-26,80,-63,93,-70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "G:=gausselim(M); # donne la matrice \351chelon; ce qui per met d'\351tudier la compatibilit\351 du syst\350me d'\351quations." }} }{PARA 0 "" 0 "" {TEXT -1 82 "Le syst\350me est compatible car on a 4 \+ variables et 4 pivots qui sont 2, 7, 195 et " }{XPPEDIT 18 0 "16008/6 5;" "6#*&\"&3g\"\"\"\"\"#l!\"\"" }{TEXT -1 70 " et par cons\351quent, une solution unique qu'on obtient avec \"backsub\" " }}{PARA 0 "" 0 " " {TEXT -1 100 "Remarque : un pivot est le premier nombre non nul sur \+ une ligne de la matrice augment\351e \351chelonn\351e. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "backsub(G); #avec la substitution \+ \340 reculons,la solution du syst\350me est :" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Plus \351l\351gamment, on aurait pu afficher : " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "[x1,x2,x3,x4]=backsub(G); " }}}{PARA 0 "" 0 "" {TEXT -1 101 "On peut faire la m\351thode de Gauss, \351tape par \351tape avec Maple : voir le laboratoire 8 GaussEtapes. mws" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 290 39 "Cas o\371 il y a une infinit\351 de solutions" } {TEXT 291 1 " " }}{PARA 0 "" 0 "" {TEXT 299 11 "Exemple 2 :" }{TEXT -1 80 " R\351soudre le syst\350me compos\351 des trois premi\350res \+ \351quations du syst\350me pr\351c\351dent." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "2*x1+3*x2+12*x3-7*x4 = 31;" "6#/,**&\"\"#\"\"\"%#x1GF'F '*&\"\"$F'%#x2GF'F'*&\"#7F'%#x3GF'F'*&\"\"(F'%#x4GF'!\"\"\"#J" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "5*x1+7*x2-4*x3+19*x4 = 3 6;" "6#/,**&\"\"&\"\"\"%#x1GF'F'*&\"\"(F'%#x2GF'F'*&\"\"%F'%#x3GF'!\" \"*&\"#>F'%#x4GF'F'\"#O" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "-8*x1+7*x2-12*x3-11*x4 = -44;" "6#/,**&\"\")\"\"\"%#x1GF'!\"\"*&\" \"(F'%#x2GF'F'*&\"#7F'%#x3GF'F)*&\"#6F'%#x4GF'F),$\"#WF)" }{TEXT -1 47 "\nqui correspondent aux 3 premi\350res lignes de M " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "M2:=delrows(M,4..6);# matrice obten ue en enlevant les lignes 4 \340 6 de la matrice M." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G2:=gausselim(M2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "backsub(%);" }}}{PARA 0 "" 0 "" {TEXT -1 125 " Le syst\350me est compatible avec 4 variables et 3 pivots. Il y a une \+ variable libre qui est x4; Maple lui donne la valeur _t1. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Donc x1 = " }{XPPEDIT 18 0 "851/157-466/157*_t[1] ;" "6#,&*&\"$^)\"\"\"\"$d\"!\"\"F&*(\"$m%F&F'F(&%#_tG6#F&F&F(" }{TEXT -1 17 " , x2 = " }{XPPEDIT 18 0 "613/314+3/157*_t[1];" "6#,&* &\"$8'\"\"\"\"$9$!\"\"F&*(\"\"$F&\"$d\"F(&%#_tG6#F&F&F&" }{TEXT -1 15 ", x3 = " }{XPPEDIT 18 0 "1497/1256+337/314*_t[1];" "6#,&*&\"% (\\\"\"\"\"\"%c7!\"\"F&*(\"$P$F&\"$9$F(&%#_tG6#F&F&F&" }{TEXT -1 25 " \+ et x4 = _t1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Une solution particuli\350re : " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "[x1,x2,x3,x4]=subs(_t[1]=1,% );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "La solution g\351n\351rale \+ peut s'\351crire : " }{XPPEDIT 18 0 "vector([851/157, 613/314, 1497/12 56, 0]);" "6#-%'vectorG6#7&*&\"$^)\"\"\"\"$d\"!\"\"*&\"$8'F)\"$9$F+*& \"%(\\\"F)\"%c7F+\"\"!" }{TEXT -1 6 " + t1*" }{XPPEDIT 18 0 "vector([- 466/157, 3/157, 337/314, 1]);" "6#-%'vectorG6#7&,$*&\"$m%\"\"\"\"$d\"! \"\"F,*&\"\"$F*F+F,*&\"$P$F*\"$9$F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 277 33 "Cas o\371 \+ il n'y a pas de solutions " }}{PARA 0 "" 0 "" {TEXT 300 11 "Exemple 3 \+ :" }{TEXT -1 21 " R\351soudre le syst\350me " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "2*x1-5*x2+6*x3 = 7;" "6#/,(*&\"\"#\"\"\"%#x1GF 'F'*&\"\"&F'%#x2GF'!\"\"*&\"\"'F'%#x3GF'F'\"\"(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "-12*x1+8*x3 = 3;" "6#/,&*&\"#7\"\"\"%#x1 GF'!\"\"*&\"\")F'%#x3GF'F'\"\"$" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "18*x1+15*x2-34*x3 = 15;" "6#/,(*&\"#=\"\"\"%#x1GF'F'*& \"#:F'%#x2GF'F'*&\"#MF'%#x3GF'!\"\"F*" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "eq1:=2*x1-5*x2+6*x3 = 7; eq2:=-12*x1+8*x3 = 3;eq3:=18*x1+15*x2-34*x3 = 15;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 76 "Cr\351ons la matrice augment\351e [A|b] avec genma trix(\351quations, variables, flag)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "genmatrix([eq1,eq2,eq3],[x1,x2,x3],flag);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gausselim(%);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 132 "Le syst\350me n'a pas de solution (\340 la troisi \350me ligne, 0x1 + 0x2 + 0x3 = 42). Si on fait backsub, Maple r\351po nd \"syst\350me inconsistant\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "backsub(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT 330 87 "Autres fa\347ons r\351soudre un syst\350me d'\351quations lin\351aires Ax = b : linsolve(A,b) et solv e" }}{PARA 0 "" 0 "" {TEXT 301 11 "Exemple 4 :" }{TEXT -1 1 " " } {TEXT 302 2 "A)" }{TEXT -1 35 " On peut chercher la solution avec " } {TEXT 257 13 "linsolve(A,b)" }{TEXT -1 1 ";" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 82 "eq1:=3*x+5*y-8*z=5;\neq2:=2*x+7*y-3*z=7;\neq3:=-5*x +2*y = 12;\neq4:=10*x+10*y-11*z=0;" }}}{PARA 0 "" 0 "" {TEXT -1 54 "Cr \351ons la matrice des coefficients A avec la commande " }{TEXT 258 9 "genmatrix" }{TEXT -1 42 ", et le vecteur-colonne des constantes b :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=genmatrix([eq1,eq2,eq3, eq4],[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b:=vector ([5,7,12,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(A ,b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 332 2 "B)" }{TEXT -1 35 " On peu t chercher la solution avec " }{TEXT 331 5 "solve" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(\{eq1,eq2,eq3,eq4\},\{ x,y,z\});" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Note : Avec la commande \+ \253" }{TEXT 289 7 "geneqns" }{TEXT -1 77 "\273 on peut transformer un e matrice augment\351e en syst\350me d'\351quations lin\351aires." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "geneqns(A,[r,s,t],b);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 0 "" }{TEXT 278 46 "Syst\350mes d'\351quations lin\351aires avec param\350tres" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 279 42 "M\351thod e avec gausselim(M) suivi de backsub" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 10 "Exemple 5)" }{TEXT -1 98 " Pour quelles valeurs de a, b \+ et c, le syst\350me d'\351quations lin\351aires suivant, est-il compa tible ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eq1:=x+3*y=a;eq2 :=3*x-y=b;eq3:=7*x+2*y=c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "M:=genmatrix([eq1,eq2,eq3],[x,y],flag);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gausselim(M);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "backsub(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 65 " Maple affirme que l e syst\350me est inconsistant, ce qui est exact " }{TEXT 263 12 "seule ment si" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c-13/10*a-19/10*b;" "6#,(%\"c G\"\"\"*(\"#8F%\"#5!\"\"%\"aGF%F)*(\"#>F%F(F)%\"bGF%F)" }{TEXT -1 7 " \+ <> 0 " }}{PARA 0 "" 0 "" {TEXT 264 54 "Le syst\350me est consistant e t a une solution unique si " }{TEXT 303 1 " " }{XPPEDIT 18 0 "c-13/10* a-19/10*b;" "6#,(%\"cG\"\"\"*(\"#8F%\"#5!\"\"%\"aGF%F)*(\"#>F%F(F)%\"b GF%F)" }{TEXT -1 5 " = 0 " }}{PARA 0 "" 0 "" {TEXT 274 17 " Maple cons id\350re " }{XPPEDIT 18 0 "c-13/10*a-19/10*b" "6#,(%\"cG\"\"\"*(\"#8F% \"#5!\"\"%\"aGF%F)*(\"#>F%F(F)%\"bGF%F)" }{TEXT 265 42 " comme une exp ression diff\351rente de z\351ro. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 18 "Attention : Maple " }{TEXT 259 9 "consid \350re" }{TEXT 260 55 " toute expression contenant des param\350tres c omme <> 0. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 " Rappel : Soit A u ne matrice 4x4 dont les colonnes sont form\351es des vecteurs v1,v2,v3 ,v4, A = [v1,v2,v3,v4] et soit x et b des vecteurs colonnes." }}{PARA 0 "" 0 "" {TEXT -1 24 "A = [v1,v2,v3,v4] , x =" }{XPPEDIT 18 0 "matri x([[x1], [x2], [x3], [x4]])" "6#-%'matrixG6#7&7#%#x1G7#%#x2G7#%#x3G7#% #x4G" }{TEXT -1 8 " et b = " }{XPPEDIT 18 0 "matrix([[b1], [b2], [b3], [b4]]);" "6#-%'matrixG6#7&7#%#b1G7#%#b2G7#%#b3G7#%#b4G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "R\351soudre le syst \350me d'\351quations lin\351aires Ax = b est \351quivalent \340 r\351 soudre le syst\350me d'\351quations lin\351aires x1v1 + " }{TEXT 293 0 "" }{TEXT -1 38 "x2v2 +x3v3 + x4v4 = b car Ax = x1v1 + " }{TEXT 294 0 "" }{TEXT -1 17 "x2v2 +x3v3 + x4v4" }}{PARA 0 "" 0 "" {TEXT -1 28 "V oir Laboratoire 4 Matrices-" }{TEXT 304 47 "Combinaison lin\351aire de s colonnes d'une matrice" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "v1:=matrix(4,1,[2,-5,6,7]):v2:=matrix(4,1,[-6,15,-18 ,-21]):v3:=matrix(4,1,[-12,0,8,3]):v4:=matrix(4,1,[18, 15, -34, -27]): b:=matrix(4,1,[b1,b2,b3,b4]):\nv1=evalm(v1),v2=evalm(v2),v3=evalm(v3), v4=evalm(v4),b=evalm(b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 10 "Exemple 6)" }{TEXT -1 17 " R \351soudre x1v1 + " }{TEXT 313 0 "" }{TEXT -1 21 "x2v2 +x3v3 + x4v4 = \+ b" }}{PARA 0 "" 0 "" {TEXT 309 77 "La solution de Ax = b en interpr \351tant la matrice \351chelonn\351e avec gausselim : " }}{PARA 0 "" 0 "" {TEXT -1 59 "La matrice augment\351e M = [A | b] = [v1,v2,v3,v4 | b] est : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "M:=augment(v1 ,v2,v3,v4,b);" }}}{PARA 0 "" 0 "" {TEXT -1 27 "La matrice \351chelonn \351e est :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gausselim(M); " }}}{PARA 0 "" 0 "" {TEXT 312 49 "La solution en interpr\351tant la m atrice \351chelonn\351e" }{TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 32 "le syst\350me est incompatible si " }{XPPEDIT 18 0 "b4+1/4*b1+3/2 *b2;" "6#,(%#b4G\"\"\"*(F%F%\"\"%!\"\"%#b1GF%F%*(\"\"$F%\"\"#F(%#b2GF% F%" }{TEXT -1 39 " <> 0 et le syst\350me est compatible si " } {XPPEDIT 18 0 "b4+1/4*b1+3/2*b2;" "6#,(%#b4G\"\"\"*(F%F%\"\"%!\"\"%#b1 GF%F%*(\"\"$F%\"\"#F(%#b2GF%F%" }{TEXT -1 80 " = 0 et alors on a une i nfinit\351 de solutions (4 variables, 2 pivots : 2 et -30)." }}{PARA 0 "" 0 "" {TEXT -1 51 "La r\351ponse de Maple si on tente de r\351soud re avec : " }{TEXT 315 62 "1) soit gausselim avec backsub, ou 2) lins olve, ou 3) solve." }}{PARA 0 "" 0 "" {TEXT 310 38 "1) Appliquons bac ksub \340 gausselim(M) :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " backsub(%);" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Ici on a une " }{TEXT 311 23 "mauvaise interpr\351tation" }{TEXT -1 56 " si on conclut que l e syst\350me est toujours incompatible " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 307 34 "2) Si on cherche la solution avec " }{TEXT -1 1 " " }{TEXT 308 13 "linsolve(A,b)" }{TEXT -1 39 ", on a la m\352me mauvaise interpr\351tation." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Soit A est \+ la matrice des coefficients et b la colonne des constantes." }{TEXT 316 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "La matrice des coefficients A est obtenue de M en \351liminant la cinqui\350me colonne, soit la col onne des constantes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A:= delcols(M,5..5);# la matrice des coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(A,b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "En n'affichant rien, Maple r\351pond que le syst\350me n'a pas de solution alors que cela est vrai seulement si " }{XPPEDIT 18 0 "b4 +1/4*b1+3/2*b2;" "6#,(%#b4G\"\"\"*(F%F%\"\"%!\"\"%#b1GF%F%*(\"\"$F%\" \"#F(%#b2GF%F%" }{TEXT -1 6 " <> 0 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 323 2 "3)" }{TEXT -1 1 " " }{TEXT 317 31 "Si on cherche la solution av ec " }{TEXT 318 5 "solve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "with(student,equate):equate(x1*v1 + x2*v2 +x3*v3 + x4*v4, b);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(%,\{x1,x2,x3,x4\});" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "En n'affichant rien, Maple r\351 pond que le syst\350me n'a pas de solution" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "Dans cet exemple, et en g\351n\351ral, la m\351thode de \+ Gauss (gausselim sans backsub) est pr\351f\351rable \340 linsolve et m \352me solve pour \351tudier la compatibilit\351 d'un syst\350me d' \351quations lin\351aires" }{TEXT 321 1 " " }{TEXT 320 15 "avec param \350tres" }{TEXT 322 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "Voir le compl\351ment 6b" }{TEXT 319 19 " AvantageGauss.mws " } {TEXT -1 22 "pour un autre exemple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 275 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 63 "Pour \351tudier la consistance d'un syst\350me \+ d'\351quations lin\351aires " }{TEXT 261 15 "avec param\350tres" } {TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT 262 46 "Trouvez la matrice \+ \351chelonn\351e avec gausselim. " }{TEXT -1 82 "Ne pas r\351soudre : \+ ni avec gausselim avec backsub, ni avec linsolve, ni avec solve." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 137 "Interpr\351tez la matric e \351chelonn\351e, (En particulier, faites une \351tude particuli\350 re pour des valeurs qui annulent un ou des d\351nominateurs)." }{TEXT 271 1 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 276 64 "Etude compl\350te d 'un syst\350me d'\351quations lin\351aires avec param\350tre" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq1:=a*x+3*y-4*z =5;eq2:=7*x -a*y-6*z=12;eq3:=3*x+4*y-a*z=5;" }}}{PARA 0 "" 0 "" {TEXT 305 10 "Exem ple 7)" }{TEXT -1 70 " Pour quelles valeurs de a, le syst\350me d'\351 quations est-il compatible ?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "M:=matrix(3,4,[a,3,-4,5,7,-a,-6,12,3,4,-a,5]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "M1:=gausselim(M);" }}}{PARA 0 "" 0 "" {TEXT -1 32 "Le syst\350me est incompatible si " }{XPPEDIT 18 0 "-(33 *a-166+a^3)/(4*a-9);" "6#,$*&,(*&\"#L\"\"\"%\"aGF(F(\"$m\"!\"\"*$F)\" \"$F(F(,&*&\"\"%F(F)F(F(\"\"*F+F+F+" }{TEXT -1 9 " = 0 et " } {XPPEDIT 18 0 "(33*a-143+5*a^2)/(4*a-9);" "6#*&,(*&\"#L\"\"\"%\"aGF'F' \"$V\"!\"\"*&\"\"&F'*$F(\"\"#F'F'F',&*&\"\"%F'F(F'F'\"\"*F*F*" }{TEXT -1 7 " <> 0. " }}{PARA 0 "" 0 "" {TEXT -1 86 "Mais qu'arrive-t-il si \+ a = 9/4 ? ou a = 0? (valeurs qui annulent des d\351nominateurs). " }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "\311tudions ces 2 cas :" } {MPLTEXT 1 0 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 41 "a) Rempla\347ons a \+ = 9/4 dans la matrice M :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "M2:=subs(a=9/4,op(M));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " gausselim(M2);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "Le" }{TEXT 266 40 " sys t\350me est donc compatible si a = 9/4" }{TEXT -1 98 " car on un syst \350me consistant avec 3 variables, 3 pivots. Par cons\351quent, la so lution unique est :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "backs ub(%);" }}}{PARA 0 "" 0 "" {TEXT -1 32 "b) Regardons maintenant a = 0 . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "M2:=subs(a=0,op(M));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gausselim(M2);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "Le" }{TEXT 268 35 " syst\350me est donc compatib le a = 0" }{TEXT -1 56 ", on a 3 variables, 3 pivots et la solution u nique est :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "backsub(%);" }}}{PARA 0 "" 0 "" {TEXT -1 74 "Rempla\347ons a = 0 dans le syst\350me d'\351quations; on a un syst\350me d'\351quations " }{TEXT 269 15 "sa ns param\350tres" }{TEXT -1 73 ". On peut r\351soudre avec n'importe q uelle m\351thode, par exemple avec solve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(a=0,\{eq1,eq2,eq3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "solve(\{3*x+4*y = 5,3*y-4*z = 5,7*x-6*z = 12\},\{x ,y,z\});" }}}{PARA 0 "" 0 "" {TEXT -1 2 "Le" }{TEXT 267 43 " syst\350m e est donc aussi compatible si a = 0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 272 51 "Le syst\350me est donc incompatible si et seulement s i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-(33*a-166+a^3)/(4*a-9);" "6#,$*&,( *&\"#L\"\"\"%\"aGF(F(\"$m\"!\"\"*$F)\"\"$F(F(,&*&\"\"%F(F)F(F(\"\"*F+F +F+" }{TEXT -1 9 " = 0 et " }{XPPEDIT 18 0 "(33*a-143+5*a^2)/(4*a-9); " "6#*&,(*&\"#L\"\"\"%\"aGF'F'\"$V\"!\"\"*&\"\"&F'*$F(\"\"#F'F'F',&*& \"\"%F'F(F'F'\"\"*F*F*" }{TEXT -1 65 " <> 0. Pour quelles valeurs de a , le syst\350me est-il incompatible?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Cherchons les z\351ros de " }{XPPEDIT 18 0 "33*a-166+a^3" "6#,(*&\"#L\"\"\"%\"aGF&F&\"$m\"!\"\"*$F'\"\"$F&" }{TEXT -1 29 " = 0. Tra\347ons son graphique." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(33*a-166+a^3,a=-infinity..infinity);" }}} {PARA 0 "" 0 "" {TEXT -1 18 "On a un seul z\351ro." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 70 "zero:=fsolve(33*a-166+a^3=0); #le seul z\351 ro de -(33*a-166+a^3)/(4*a-9)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(a=zero,33*a-143+5*a^2);#a n'est pas un z\351ro de (33*a-143 +5*a^2)/(4*a-9)" }}}{PARA 0 "" 0 "" {TEXT -1 15 "Ainsi, si a = " } {MPLTEXT 0 21 100 "3.607559577;le syst\350me est incompatible car -(33 *a-166+a^3)/(4*a-9)=0 et (33*a-143+5*a^2)/(4*a-9)<>0" }}{PARA 0 "" 0 " " {TEXT -1 52 "Le syst\350me est donc incompatible seulement pour a = \+ " }{XPPEDIT 18 0 "3.607559577;" "6#$\"+x&fvg$!\"*" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 34 "Syst\350mes d'\351quations non lin\351aires" }{TEXT 324 0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "La m\351 thode de Gauss n'est valide que pour les syst\350mes d'\351quations li n\351aires. " }{TEXT 327 17 "Dans certains cas" }{TEXT -1 45 ", on peu t transformer un syst\350me d'\351quations " }{TEXT 325 13 "non lin \351aires" }{TEXT -1 153 " en un syst\350me d'\351quations lin\351aire s, que l'on peut r\351soudre avec une m\351thode de r\351solution de s yst\350mes d'\351quations lin\351aires comme la m\351thode de Gauss. \+ " }}{PARA 0 "" 0 "" {TEXT 329 10 "Exemple 8)" }{TEXT -1 33 " r\351soud re le syst\350me d'\351quations " }{TEXT 326 13 "non lin\351aires" } {TEXT -1 9 " suivant:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "eq1:=26*x^2+12*y^2+40*z^2=525;eq2:= 64*x^2-81*y^2+z^2=220;eq3:=32*x^2+45*y^2-23*z^2=234;" }}}{PARA 0 "" 0 "" {TEXT -1 86 "Rempla\347ons x^2,y^2 et z^2 respectivement par u, v, \+ w par les substitutions suivantes: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "nouveau:=subs(\{x^2=u,y^2=v,z^2=w\},[eq1,eq2,eq3]);" }}}{PARA 0 "" 0 "" {TEXT -1 39 "On a maintenant un syst\350me d'\351qu ations " }{TEXT 328 9 "lin\351aires" }{TEXT -1 64 ". R\351solvons avec la m\351thode de Gauss. La matrice augment\351e est :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "M:=genmatrix([nouveau[1],nouveau[2] ,nouveau[3]],[u,v,w],flag);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "backsub(gausselim(M));" }}}{PARA 0 "" 0 "" {TEXT -1 22 "On a donc \+ x^2 = " }{XPPEDIT 18 0 "362023/47366;" "6#*&\"'B?O\"\"\"\"&mt%! \"\"" }{TEXT -1 25 ", y^2 = " }{XPPEDIT 18 0 "80783/2 3683;" "6#*&\"&$y!)\"\"\"\"&$oB!\"\"" }{TEXT -1 22 " et z ^2= " }{XPPEDIT 18 0 "168947/23683;" "6#*&\"'Z*o\"\"\"\"\"&$oB!\"\"" } }{PARA 0 "" 0 "" {TEXT -1 30 "On obtient x = +/-sqrt( " } {XPPEDIT 18 0 "362023/47366" "6#*&\"'B?O\"\"\"\"&mt%!\"\"" }{TEXT -1 23 "), y = +/-sqrt( " }{XPPEDIT 18 0 "80783/23683;" "6#*&\"&$y! )\"\"\"\"&$oB!\"\"" }{TEXT -1 23 ") et z = +/-sqrt( " }{XPPEDIT 18 0 "168947/23683;" "6#*&\"'Z*o\"\"\"\"\"&$oB!\"\"" }{TEXT -1 3 "). \+ " }}{PARA 0 "" 0 "" {TEXT -1 147 "Ce qui donne 8 triplets solutions (x ,y,z), qu'on peut obtenir par Maple avec solve. Note : dans les cas d' une \351quation polyn\364miale, utiliser fsolve" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "solve(\{x^2= 362023/47366,y^2=80783/23683 ,z^2 = 168947/23683\},\{x,y,z\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "allvalues(%); #allvalues donne les solutions de RootOf( )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(%);# evalf pour avo ir les valeurs en d\351cimales" }}}{PARA 0 "" 0 "" {TEXT -1 121 "Dans \+ ce probl\350me, il aurait \351t\351 plus facile d'utiliser au d\351but la commande solve. On peut \351crire en une seule commande :" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "evalf(allvalues(solve(\{eq1, eq2,eq3\},\{x,y,z\})));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT 257 9 "Exercices" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 333 5 "No 1)" }{TEXT -1 184 " R\351soudre le sys t\350me d'\351quations lin\351aires par la m\351thode d'\351limination de Gauss.\n 2x - 2y - 4z = -2\011\011\011\n 2x - y - z = 2\011 \011\011 \n -3x + 5y + 4z = 3\011\011\011 \n -x - 5y + 4 z = -3" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 288 32 " Espace de travail d e 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 334 5 "No 2)" } {TEXT -1 80 " R\351soudre le syst\350me d'\351quations lin\351aires pa r la m\351thode d'\351limination de Gauss" }}{PARA 0 "" 0 "" {TEXT -1 74 "2x1 - 6x2 + 3x3 - 2x4 = -1\n -x1 + 3x2 - 2x3 = 4\n 3x1 - 9x2 + 4x3 - 4x4 = 1" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 287 31 "E space 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 335 5 "No 3)" }{TEXT -1 56 " R\351soudre le syst\350me d'\351qua tions homog\350nes avec linsolve" }}{PARA 0 "" 0 "" {TEXT -1 17 "-x + \+ 6y + 15z = 0" }}{PARA 0 "" 0 "" {TEXT -1 18 "2x - 12y - 30z = 0" }} {PARA 0 "" 0 "" {TEXT -1 16 "7x + 5y -11z = 0" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 286 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 336 5 "No 4)" }{TEXT -1 230 " Pour que lles valeurs de a et b le syst\350me d'\351quations suivant a-t-il \n \+ a) une solution unique ?\n b) pas de solution ?\n \+ c) une infinit\351 de solutions ?\nx + 2y + az = 4\n2x - y + 3z = b\n3x - 4y + 2z = -3" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 285 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 337 5 "No 5)" }{TEXT -1 97 " Pour quelles valeurs de a et b le syst\350me d'\351quations suivant a-t-il une infinit\351 de soluti ons ?" }}{PARA 0 "" 0 "" {TEXT -1 16 "3x - 5y + az = 4" }}{PARA 0 "" 0 "" {TEXT -1 17 "-2x + 2y - 5z = 2" }}{PARA 0 "" 0 "" {TEXT -1 16 "7x + 2y - 6z = b" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 32 " Espace de tr avail 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 338 5 "No 6)" } {TEXT -1 102 " \300 l'aide de la m\351thode de Gauss et de la m\351tho de linsolve, r\351soudre le syst\350me d'\351quations Ax = b o\371 " } }{PARA 0 "" 0 "" {TEXT -1 4 "A = " }{XPPEDIT 18 0 "matrix([[-1, 6, 15] , [2, -12, -30], [7, 5, -11]]);" "6#-%'matrixG6#7%7%,$\"\"\"!\"\"\"\"' \"#:7%\"\"#,$\"#7F*,$\"#IF*7%\"\"(\"\"&,$\"#6F*" }}{PARA 0 "" 0 "" {TEXT -1 7 "et b = " }{XPPEDIT 18 0 "matrix([[-17], [34], [25]])" "6#- %'matrixG6#7%7#,$\"# " 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 339 5 "No 7)" }{TEXT -1 36 " M\352me question que 6) mais avec b = " }{XPPEDIT 18 0 "matrix([[12], [5], [-7]]);" "6#-%'matrixG6#7%7#\"#77#\"\"&7#,$\"\"(! \"\"" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 282 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 340 6 "No \+ 8) " }{TEXT 341 49 "Soit (-2,-3), (4,7), (5,-6) et (-4,5), 4 points. \+ " }}{PARA 0 "" 0 "" {TEXT 342 121 "a) Trouver a,b,c,d pour que la cour be de y = a*x^3 + b*x^2 + c*x + d. \nb) Trouver les z\351ros de la fo nction obtenue en a)" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 281 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 343 5 "No 9)" }{TEXT -1 65 " a) Rendre lin\351aire le syst \350me d'\351quations non lin\351aires suivant." }}{PARA 0 "" 0 "" {TEXT -1 23 " 4e^x + 5e^y =10" }}{PARA 0 "" 0 "" {TEXT -1 21 " \+ -3e^x+7e^y = 2" }}{PARA 0 "" 0 "" {TEXT -1 88 " b) R\351soudre , par la m\351thode de Gauss, le syst\350me d'\351quations lin\351aire s obtenu en a) ." }}{PARA 0 "" 0 "" {TEXT -1 60 " c) Donner la solut ion du syst\350me d'\351quations donn\351 en a)." }}{PARA 0 "" 0 "" {TEXT -1 101 " d) V\351rifiez votre r\351ponse en r\351solvant les \+ \351quations donn\351es en a) avec la commande Maple : solve." }} {PARA 0 "" 0 "" {TEXT -1 36 "Note : e^x s'\351crit exp(x), en Maple " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 280 31 "Espace de tra vail 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 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }