12-Equa\303\247\303\265es Lineares 2 x 2 Solucionando sistemas de equa\303\247\303\265es lineares 2 x 2 usando tr\303\252s m\303\251todos - gr\303\241fico, substitui\303\247\303\243o e elimina\303\247\303\243o

restart; with(plots):

Warning, the name changecoords has been redefined

Gr\303\241fico Sistema := proc( eqns ) local eq1, eq2, sys,S,soln, x1,y1,xl,xr,L1,L2,H,V,ST,Point,wid,m1,m2; eq1 := eqns[1]: eq2 := eqns[2]: #------- check for solubility ---------------- m1 := subs(x = 1,solve(eq1,y))- subs(x = 0,solve(eq1,y)); m2 := subs(x = 1,solve(eq2,y))- subs(x = 0,solve(eq2,y)); if(m1<>m2) #------- solve the system ---------------- then sys := {eq1, eq2}; y1 := solve( eq1, y); subs(y = y1, eq2); x1 := solve( %, x); y1 := subs( x = x1, y1); soln := [x1, y1]; xl := min( -1, 1.5*x1 ); xr := max( 1, 1.5*x1); wid := ((xr-xl)/20); #------- create plots ---------------- Point := plottools[disk]([ x1, y1], wid, color=red ): H := plot( [[ 0,y1],[x1,y1]], color = orange, linestyle = 2 ); V := plot( [[x1,0], [x1,y1]], color = gold, linestyle = 2 ); ST := textplot( [ x1+1,y1-1, convert(soln,string) ] ); L1 := plot( solve( eq1, y), x = xl..xr,thickness = 2, color=blue ): L2 := plot( solve( eq2, y), x = xl..xr,thickness = 2, color=green ): plots[display]( L1, L2, Point, H, V, ST, scaling=constrained ); #------- Display Result ---------------- else xl:=-5; xr:=5; L1 := plot( solve( eq1, y), x = xl..xr,thickness = 2, color=blue ): L2 := plot( solve( eq2, y)+.08, x = xl..xr,thickness = 2, color=green ): plots[display]( L1, L2, scaling=constrained); fi; end proc: Eliminate := proc( sys) print(`Adicione essas duas equa\303\247\303\265es:`); print(sys[1]); print(sys[2]); print(`___________`); sys[1]+sys[2]; end proc:

Um m\303\251todo b\303\241sico, por\303\251m perspicaz, de se resolver sistemas de equa\303\247\303\265es \303\251 por examinando o seu lado gr\303\241fico. sys:= { 12*x + 13*y = 11, 7*x - 6*y = 20}; SysPlot ( sys ); solve( sys, {x,y} );

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sys:= { 2*x + 5*y = -7, 3*x - 6*y = -24}; solve( sys, {x,y} ); SysPlot ( sys );

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sys:= { 2*x + 7*y = 28, 3*x - 5*y = 11}; solve( sys, {x,y} ); SysPlot ( sys );

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Nos exemplos acima, podemos encontrar uma solu\303\247\303\243o. Geralmente, quando queremos resolver um sistema, nos deparamos com tr\303\252s possibilidades: 1. O sistema \303\251 independente e consistente - neste caso existe apenas uma solu\303\247\303\243o 2. O sistema \303\251 inconsistente - este caso, n\303\243o existe solu\303\247\303\243o 3. O sistema \303\251 dependente - este caso h\303\241 um n\303\272mero infinito de solu\303\247\303\265es. Vejamos o que acontece quando encontramos os dois \303\272ltimos casos de sistemas que n\303\243o produzem uma \303\272nica solu\303\247\303\243o. I. Sistemas Independentes & ConsistentesAqui temos um sistema independente e consistente. \303\211 consistente porque as linhas se intersecionam e as linhas s\303\243o independentes - significando que n\303\243o s\303\243o a mesma linha. Quaisquer duas linhas n\303\243o paralelas s\303\243o consistentes. sys:= { 2*x + 5*y = 28, 3*x - 6*y = 15}; solve( sys, {x,y} ); SysPlot ( sys );

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sys:= { 8*x + 3*y = 14, 11*x + 2*y = -2}; solve( sys, {x,y} ); SysPlot ( sys );

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II. Sistemas InconsistentesO sistema n\303\243o \303\251 consistente porque suas linhas n\303\243o se intersecionam. As linhas s\303\243o paralelas e Maple n\303\243o \303\251 capaz de resolver um sistema que n\303\243o tem solu\303\247\303\243o. sys:= { 3*x + 7*y = 42, 3*x + 7*y = 0 }; SysPlot ( sys );

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NiMiIiEtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYlUSJ4NiJRITYiLSUlRk9OVEc2JSUoREVGQVVMVEclKERFRkFVTFRHIiM3LSUoU0NBTElOR0c2IyUsQ09OU1RSQUlORURHLSUpX1ZJU0lCTEVHNiMiIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRJIiEiIi0lKUJPVU5EU19ZRzYjJCIjISkhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUhWyQhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlU04hIiItJSlDSElMRFJFTkc2Ii0lK0FOTk9UQVRJT05HNictJSlCT1VORFNfWEc2IyQiIiEhIiItJSlCT1VORFNfWUc2IyQiIiEhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUrUyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrUyEiIi0lKUNISUxEUkVORzYiRzYi

sys:= { 10*x - 2*y = 40, 10*x - 2*y = 20 }; SysPlot ( sys );

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

III. Sistemas Dependentes Se duas linhas forem exatamente a mesma linha, s\303\243o chamadas dependentes . H\303\241 apenas uma linha no sistema, pois as duas linhas s\303\243o, na verdade, a mesma. Nesse caso, TODOS os pontos s\303\243o pontos de interse\303\247\303\243o. H\303\241 um n\303\272mero infinito de solu\303\247\303\265es - todos os pontos da linha. sys := { 4*x - 6*y = 30, 3*y = 2*x-15 }; SysPlot ( sys );

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NiI=

sys := { y = -2*x + 12, 2*y = 24 -4*x }; SysPlot ( sys );

NiM+JSRzeXNHPCQvJSJ5RywmKiYiIiMiIiIlInhHIiIiISIiIiM3IiIiLywkKiYiIiMiIiIlInlHIiIiIiIiLCYiI0MiIiIqJiIiJSIiIiUieEciIiIhIiI=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NiI=

Problema 3.1 : 3x + 8y = -1 5x - 2y = 29 0. Escreva o sistema sys := { 3*x + 8*y = -1 , 5*x - 2*y = 29 }: 1. Solucione uma das equa\303\247\303\265es para uma das vari\303\241veis. H\303\241 quatro op\303\247\303\265es, mas escolha a que parecer mais f\303\241cil. Nesse caso todas elas parecem ter praticamente o mesmo n\303\255vel de dificuldade. Vamos resolver para y na primeira equa\303\247\303\243o. Lembre-se de qual y ser\303\241 representado em termos de n\303\272meros e x. y1 := solve( sys[1], y );

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2. Agora, substitua ESSE y, pelo y na OUTRA equa\303\247\303\243o. Esta \303\251 a chave desse m\303\251todo. Uma vez que fazemos isso, temos uma equa\303\247\303\243o - mas apenas uma vari\303\241vel x. Essa \303\251 uma equa\303\247\303\243o que podemos resolver. subs(y = y1, sys[2]);

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3. Solucione essa equa\303\247\303\243o para x x1 := solve( %, x);

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4. Substitua esse valor de x em uma das equa\303\247\303\265es, para encontrar y. y1 := subs( x = x1, y1);

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Esta \303\251 a solu\303\247\303\243o ! soln := [x1, y1];

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Podemos tirar a prova substituindo o valor nas duas equa\303\247\303\265es. Veja que ambas as solu\303\247\303\265es s\303\243o corretas j\303\241 que -1 = -1. Pronto. subs( {x = x1, y = y1}, {sys[1], sys[2]});

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Usando o Maple, temos outra maneira de provar a resposta - \303\251 s\303\263 solucionarmos usando um comando do Maple . solve( sys, {x,y} );

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Vamos tentar outros exemplos. Problema 3.2 : Solucione o sistema : 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 Se somarmos essas equa\303\247\303\265es, nenhuma das vari\303\241veis ser\303\241 cancelada. sys := { 7*x + 2*y = 8 , 5*x - 2*y = 16 }: y1 := solve( sys[1], y);

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subs(y = y1, sys[2]);

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x1 := solve( %, x);

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y1 := subs( x = x1, y1);

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soln := [x1, y1];

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Podemos verificar se a solu\303\247\303\243o est\303\241 correta substitindo o resultado na duas equa\303\247\303\265es ou usando o comando do Maple. subs( {x = x1, y = y1}, {sys[1], sys[2]});

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solve( sys, {x,y} );

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Vamos tentar outro exemplo. Problema 3.3 : Solucione o sistema : 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 Se somarmos essas equa\303\247\303\265es, nenhuma vari\303\241vel ser\303\241 cancelada. sys := { 11*x + 6*y = 10 , 5*x - 3*y = 16 }: y1 := solve( sys[1], y);

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subs(y = y1, sys[2]);

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x1 := solve( %, x);

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y1 := subs( x = x1, y1);

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soln := [x1, y1];

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Verifique : solve( sys, {x,y} );

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E se tivermos um sistema dependente? Problema 4.6 : Solucione o sistema : 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 sys := [ 15*x - 10*y = 45, -18*x + 12*y = -54 ]: y1 := solve( sys[1], y);

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subs(y = y1, sys[2]);

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N\303\263s n\303\243o podemos fazer nada com essa equa\303\247\303\243o resultante, pois ela n\303\243o tem vari\303\241veis. Entretanto, essa equa\303\247\303\243o \303\251 verdadeira. Isso significa que a nossa suposi\303\247\303\243o de que as linhas t\303\252m um ponto de interse\303\247\303\243o \303\251 verdadeira... verdadeira para todos os valores de x. Logo, esse \303\251 um sistema dependente e a solu\303\247\303\243o consiste em todos os pontos da linha. Se tentarmos o m\303\251todo de substitui\303\247\303\243o em um sistema inconsistente, teremos : Problema 4.7 : Solucione o Sistema: 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 sys := [ 15*x - 10*y = 45, -18*x + 12*y = 54 ]: y1 := solve( sys[1], y);

NiM+JSN5MUcsJiooIiIkIiIiIiIjISIiJSJ4RyIiIiIiIiMiIioiIiMhIiI=

subs(y = y1, sys[2]);

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Claramente, h\303\241 algo errado. N\303\243o podemos resolver por nenhuma vari\303\241vel, portanto a equa\303\247\303\243o n\303\243o \303\251 verdadeira. Deve ser o caso onde n\303\243o h\303\241 um ponto de interse\303\247\303\243o. Em outras palavras, temos um sistema inconsistente - linhas paralelas, sem interse\303\247\303\243o - ent\303\243o, a resposta para esse problema \303\251 um conjunto vazio. N\303\243o h\303\241 ponto de interse\303\247\303\243o.

Certas vezes damos sorte. Vamos somar essas duas equa\303\247\303\265es. Problema 4.1 : Solucione o sistema : 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 sys := { 7*x + 2*y = 8 , 5*x - 2*y = 16 }: Eliminate( [sys[1], sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIiKCIiIiUieEciIiIiIiIqJiIiIyIiIiUieUciIiIiIiIiIik=NiMvLCYqJiIiJiIiIiUieEciIiIiIiIqJiIiIyIiIiUieUciIiIhIiIiIzs=NiMlLF9fX19fX19fX19fRw==NiMvLCQqJiIjNyIiIiUieEciIiIiIiIiI0M=

x1 := solve( %, x);

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subs(x = x1, sys[1]);

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y1 := solve( %, y); [x1, y1]; solve( sys, {x,y});

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Isso funcionou bem porque tivemos a sorte de dois dos termos terem sido cancelados: 2y e -2y. Mas, e se n\303\243o tivermos tanta sorte? A intelig\303\252ncia pode fazer a nossa sorte. Problema 4.2 : Solucione esse sistema : 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 Se somarmos as duas equa\303\247\303\265es, nenhuma das vari\303\241veis ser\303\241 cancelada. sys := { 11*x + 6*y = 10 , 5*x - 3*y = 16 }: Eliminate( [sys[1], sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIjNiIiIiUieEciIiIiIiIqJiIiJyIiIiUieUciIiIiIiIiIzU=NiMvLCYqJiIiJiIiIiUieEciIiIiIiIqJiIiJCIiIiUieUciIiIhIiIiIzs=NiMlLF9fX19fX19fX19fRw==NiMvLCYqJiIjOyIiIiUieEciIiIiIiIqJiIiJCIiIiUieUciIiIiIiIiI0U=

Entretanto, se somarmos a primeira equa\303\247\303\243o a DUAS VEZES a segunda equa\303\247\303\243o, funcionar\303\241 bem melhor. Eliminate( [sys[1], 2*sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIjNiIiIiUieEciIiIiIiIqJiIiJyIiIiUieUciIiIiIiIiIzU=NiMvLCYqJiIjNSIiIiUieEciIiIiIiIqJiIiJyIiIiUieUciIiIhIiIiI0s=NiMlLF9fX19fX19fX19fRw==NiMvLCQqJiIjQCIiIiUieEciIiIiIiIiI1U=

x1 := solve( %, x);

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subs(x = x1, sys[1]);

NiMvLCYiI0EiIiIqJiIiJyIiIiUieUciIiIiIiIiIzU=

y1 := solve( %, y); [x1, y1]; solve( sys, {x,y});

NiM+JSN5MUchIiM=NiM3JCIiIyEiIw==NiM8JC8lInlHISIjLyUieEciIiM=

Por que podemos multiplicar uma equa\303\247\303\243o por uma constante? N\303\263s fizemos exatamente a mesma coisa antes. Quando queremos solucionar 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, multiplicamos os dois lados por 7LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIzEzRidGPkY+RitGPg==. Na verdade, n\303\263s adicionamos n\303\272meros aos dois lados de uma equa\303\247\303\243o, subtra\303\255mos n\303\255meros dos dois lados de uma equa\303\247\303\243o, multiplicamos os dois lados e dividimos os dois lados. Ent\303\243o temos um longo hist\303\263rico de realizarmos essa mesma opera\303\247\303\243o aos dois lados de uma equa\303\247\303\243o e sabemos que isso n\303\243o vai mudar a natureza ou a qualidade da equa\303\247\303\243o. Agora, estamos fazendo isso para que duas vari\303\241veis sejam canceladas. Precisamos encontrar um n\303\272mero para multiplicar os dois lados da equa\303\247\303\243o, de forma que uma das vari\303\241veis se cancele. Neste caso, n\303\263s tivemos sorte pois 2 cabem em 6. Temos mais um exemplo. Dessa vez, eliminaremos x. Problema 4.3 : Solucione o sistema : 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNihGKy1GIzYnLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSgmbWludXM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlMtRiM2Ji1JI21uR0YkNiRRIjhGJ0ZALUY9Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGam4tRiw2JVEieUYnRi9GMkZARitGQC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmNvLUYjNiUtRj02LVEqJnVtaW51czA7RidGQEZCRkVGR0ZJRktGTUZPRlFGVC1GWTYkUSI2RidGQEZARitGQEYrRkA= Vamos somar as duas equa\303\247\303\265es. sys := { 7*x - 9*y = 23 , x + 8*y = -6 }: Eliminate( [sys[1], -7*sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIiKCIiIiUieEciIiIiIiIqJiIiKiIiIiUieUciIiIhIiIiI0I=NiMvLCYqJiIiKCIiIiUieEciIiIhIiIqJiIjYyIiIiUieUciIiIhIiIiI1U=NiMlLF9fX19fX19fX19fRw==NiMvLCQqJiIjbCIiIiUieUciIiIhIiIiI2w=

y1 := solve( %, y);

NiM+JSN5MUchIiI=

subs(y = y1, sys[1]);

NiMvLCYqJiIiKCIiIiUieEciIiIiIiIiIioiIiIiI0I=

x1 := solve( %, x); [x1, y1]; solve( sys, {x,y});

NiM+JSN4MUciIiM=NiM3JCIiIyEiIg==NiM8JC8lInhHIiIjLyUieUchIiI=

E se ficarmos completamente sem sorte? Ent\303\243o teremos que fabricar a sorte. Se n\303\243o houver nenhum inteiro podemos multiplicar para fazer a vari\303\241vel ser cancelada e ent\303\243o multiplicamos as duas equa\303\247\303\265es para transform\303\241-las em equa\303\247\303\265es que tenham coeficientes sim\303\251tricos em uma das vari\303\241veis. Problema 4.4 : Solucione o Sistema : LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNidGKy1GIzYoRistRiM2Ji1JI21uR0YkNiRRIjNGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGPy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWLUYsNiVRInhGJ0YvRjJGPy1GQjYtUSIrRidGP0ZFRkhGSkZMRk5GUEZSL0ZVUSwwLjIyMjIyMjJlbUYnL0ZYRmpuLUYjNiYtRjw2JFEiN0YnRj9GQS1GLDYlUSJ5RidGL0YyRj9GK0Y/LUZCNi1RIj1GJ0Y/RkVGSEZKRkxGTkZQRlIvRlVRLDAuMjc3Nzc3OGVtRicvRlhGaG8tRjw2JFEiMUYnRj9GP0YrRj8= 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 sys := { 3*x + 7*y = 1 , 4*x - 5*y = 30 }: Primeiro vamos decidir qual vari\303\241vel queremos eliminar. Vamos escolher x. Na primeira equa\303\247\303\243o, temos 3x e na segunda equa\303\247\303\243o temos 4x. Qual \303\251 o menor n\303\272mero que comporta 3 e 4 ? Essa parece uma situa\303\247\303\243o familiar. lcm(3,4);

NiMiIzc=

Ent\303\243o vamos precisar multiplicar a primeira equa\303\247\303\243o por 12/3 = 4 e a segunda por 12/4 = 3. Eliminate( [4*sys[1], 3*sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIjNyIiIiUieEciIiIiIiIqJiIjRyIiIiUieUciIiIiIiIiIiU=NiMvLCYqJiIjNyIiIiUieEciIiIiIiIqJiIjOiIiIiUieUciIiIhIiIiIyEqNiMlLF9fX19fX19fX19fRw==NiMvLCYqJiIjQyIiIiUieEciIiIiIiIqJiIjOCIiIiUieUciIiIiIiIiIyUq

Oops ! Isso n\303\243o funcionou! Ainda temos as duas vari\303\241veis! Queremos que essas equa\303\247\303\265es multiplicadas os seus coeficientes de x, ent\303\243o vamos multiplicar o segundo por -3 em vez de +3. Ent\303\243o teremos +12x e -12x, que podem ser cancelados. Eliminate( [4*sys[1], -3*sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIjNyIiIiUieEciIiIiIiIqJiIjRyIiIiUieUciIiIiIiIiIiU=NiMvLCYqJiIjNyIiIiUieEciIiIhIiIqJiIjOiIiIiUieUciIiIiIiIhIyEqNiMlLF9fX19fX19fX19fRw==NiMvLCQqJiIjViIiIiUieUciIiIiIiIhIycp

y1 := solve( %, y);

NiM+JSN5MUchIiM=

subs(y = y1, sys[1]);

NiMvLCYqJiIiJCIiIiUieEciIiIiIiIiIzkhIiIiIiI=

x1 := solve( %, x); [x1, y1]; solve( sys, {x,y});

NiM+JSN4MUciIiY=NiM3JCIiJiEiIw==NiM8JC8lInlHISIjLyUieEciIiY=

Temos outro exemplo : Problema 4.5 : Solucione o Sistema : 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNidGKy1GIzYoRistRiM2Ji1JI21uR0YkNiRRIjVGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGPy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWLUYsNiVRInhGJ0YvRjJGPy1GQjYtUSgmbWludXM7RidGP0ZFRkhGSkZMRk5GUEZSL0ZVUSwwLjIyMjIyMjJlbUYnL0ZYRmpuLUYjNiYtRjw2JFEiMkYnRj9GQS1GLDYlUSJ5RidGL0YyRj9GK0Y/LUZCNi1RIj1GJ0Y/RkVGSEZKRkxGTkZQRlIvRlVRLDAuMjc3Nzc3OGVtRicvRlhGaG8tRjw2JFEjMTZGJ0Y/Rj9GK0Y/ Vamos eliminar y : sys := { 7*x + 6*y = 9 , 5*x + 8*y = 25 }: lcm(6,8); %/6; %%/8;

NiMiI0M=NiMiIiU=NiMiIiQ=

Eliminate( 4*[sys[1], 1*sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIjRyIiIiUieEciIiIiIiIqJiIjQyIiIiUieUciIiIiIiIiI08=NiMvLCYqJiIjPyIiIiUieEciIiIiIiIqJiIjSyIiIiUieUciIiIiIiIiJCsiNiMlLF9fX19fX19fX19fRw==NiMvLCYqJiIjWyIiIiUieEciIiIiIiIqJiIjYyIiIiUieUciIiIiIiIiJE8i

x1 := solve( %, x);

NiM+JSN4MUcsJiooIiIoIiIiIiInISIiJSJ5RyIiIiEiIiMiIzwiIiciIiI=

subs(x = x1, sys[1]);

NiMvLCYqKCIjOCIiIiIiJyEiIiUieUciIiIhIiIjIiQ+IiIiJyIiIiIiKg==

y1 := solve( %, y); [x1, y1]; solve( sys, {x,y});

NiM+JSN5MUciIiY=NiM3JCwmKigiIigiIiIiIichIiIlInlHIiIiISIiIyIjPCIiJyIiIiIiJg==NiM8JC8lInhHISIkLyUieUciIiY=

E se tivermos um sistema dependente? Problema 4.6 : Solucione o Sistema : 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 sys := [ 15*x - 10*y = 45, -18*x + 12*y = -54 ]: lcm( -10, 12);

NiMiI2c=

Eliminate( [6*sys[1], 5*sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIjISoiIiIlInhHIiIiIiIiKiYiI2ciIiIlInlHIiIiISIiIiRxIw==NiMvLCYqJiIjISoiIiIlInhHIiIiISIiKiYiI2ciIiIlInlHIiIiIiIiISRxIw==NiMlLF9fX19fX19fX19fRw==NiMvIiIhIiIh

Parece que se chegou o fim da linha. N\303\243o podemos resolver essa equa\303\247\303\243o resultante nem para x nem para y. Entretanto, essa equa\303\247\303\243o \303\251 verdadeira. Isso significa que as linhas tem um ponto de interse\303\247\303\243o e a solu\303\247\303\243o \303\251 verdadeira para todos os valores de x. Logo, esse sistema \303\251 dependente e a solu\303\247\303\243o consiste em todos os pontos da linha. Se tentarmos o m\303\251todo de elimin\303\247\303\243o em um sistema inconsistente, teremos : Problema 4.7 : Solucione o Sistema : 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 sys := [ 15*x - 10*y = 45, -18*x + 12*y = 54 ]: lcm( -10, 12);

NiMiI2c=

Eliminate( [6*sys[1], 5*sys[2]] );

NiMlQkFkaWNpb25lfmVzc2FzfmR1YXN+ZXF1YXxqeXxqdnxodnxqeXxqdnxodmVzOkc=NiMvLCYqJiIjISoiIiIlInhHIiIiIiIiKiYiI2ciIiIlInlHIiIiISIiIiRxIw==NiMvLCYqJiIjISoiIiIlInhHIiIiISIiKiYiI2ciIiIlInlHIiIiIiIiIiRxIw==NiMlLF9fX19fX19fX19fRw==NiMvIiIhIiRTJg==

Claramente, algo est\303\241 errado. N\303\243o podemos resolver por essa vari\303\241vel e agora temos uma equa\303\247\303\243o falsa. Deve ser o caso onde n\303\243o h\303\241 um ponto de interse\303\247\303\243o. Em outras palavras, temos um sistema inconsistente - linhas paralelas sem interse\303\247\303\243o - logo, a resposta para o problema \303\251 um conjunto vazio. N\303\243o h\303\241 ponto de interse\303\247\303\243o.