Resolva o sistema linear [[2,I_1,-,I_2,+,3,I_3,+,4,I_4,=,9],[I_1,-,2,I_3,+,7,I_4,=,11],[[[3,I_1,-,3,I_2,+,I_3,+,5,I_4,=,8],[2,I_1,+,I_2,+,4,I_3,+,4,I_4,=,10]]]] por qualquer método.

Passo 1

Podemos resolver esse sistema pelo método da eliminação. Primeiramente, para facilitar, vamos abrir o sistema numa matriz aumentada:

([[[[2],[1],[[[3],[2]]]],[[-,1],[0],[[[-,3],[1]]]],[[[[3],[-,2],[[[1],[4]]]],[[4],[7],[[[5],[4]]]],[[9],[11],[[[8],[10]]]]]]]])

Passo 2

Para essas operações, utilizaremos o rótulo L para designar a linha e o operdor:

underset(L q u e s e r á a l t e r a d a -> O p e r a ç ã o)(->)

([[[[2],[1],[[[3],[2]]]],[[-,1],[0],[[[-,3],[1]]]],[[[[3],[-,2],[[[1],[4]]]],[[4],[7],[[[5],[4]]]],[[9],[11],[[[8],[10]]]]]]]]) underset([[L 1 -> L 1 - 2 L 2],[L 3 -> L 3 - 3 L 2],[L 4 -> L 4 - 2 L 2]])(->) ([[[[0],[1],[[[0],[0]]]],[[-,1],[0],[[[-,3],[1]]]],[[[[7],[-,2],[[[7],[8]]]],[[10],[7],[[[-,16],[-,10]]]],[[-,13],[11],[[[-,25],[-,12]]]]]]]])

Passo 3

([[[[0],[1],[[[0],[0]]]],[[-,1],[0],[[[-,3],[1]]]],[[[[7],[-,2],[[[7],[8]]]],[[10],[7],[[[-,16],[-,10]]]],[[-,13],[11],[[[-,25],[-,12]]]]]]]]) underset([[L 1 -> L 1 + L 4],[L 3 -> L 3 + 3 L 4]])(->) ([[[[0],[1],[[[0],[0]]]],[[0],[0],[[[0],[1]]]],[[[[15],[-,2],[[[31],[8]]]],[[-,20],[7],[[[-,46],[-,10]]]],[[-,25],[11],[[[-,61],[-,12]]]]]]]])

([[[[0],[1],[[[0],[0]]]],[[0],[0],[[[0],[1]]]],[[[[15],[-,2],[[[31],[8]]]],[[-,20],[7],[[[-,46],[-,10]]]],[[-,25],[11],[[[-,61],[-,12]]]]]]]]) underset(L 1 -> (1)/(15) L 1)(->) ([[[[0],[1],[[[0],[0]]]],[[0],[0],[[[0],[1]]]],[[[[1],[-,2],[[[31],[8]]]],[[-,4,//,3],[7],[[[-,46],[-,10]]]],[[-,5,//,3],[11],[[[-,61],[-,12]]]]]]]])

Passo 4

([[[[0],[1],[[[0],[0]]]],[[0],[0],[[[0],[1]]]],[[[[1],[-,2],[[[31],[8]]]],[[-,4,//,3],[7],[[[-,46],[-,10]]]],[[-,5,//,3],[11],[[[-,61],[-,12]]]]]]]]) underset([[L 2 -> L 2 + 2 L 1],[L 3 -> L 3 - 31 L 1],[L 4 -> L 4 - 8 L 1]])(->) ([[[[0],[1],[[[0],[0]]]],[[0],[0],[[[0],[1]]]],[[[[1],[0],[[[0],[0]]]],[[-,4,//,3],[13,//,3],[[[-,14,//,3],[2,//,3]]]],[[-,5,//,3],[23,//,3],[[[-,28,//,3],[4,//,3]]]]]]]])

([[[[0],[1],[[[0],[0]]]],[[0],[0],[[[0],[1]]]],[[[[1],[0],[[[0],[0]]]],[[-,4,//,3],[13,//,3],[[[-,14,//,3],[2,//,3]]]],[[-,5,//,3],[23,//,3],[[[-,28,//,3],[4,//,3]]]]]]]]) underset(A -> 3 A)(->) ([[[[0],[3],[[[0],[0]]]],[[0],[0],[[[0],[3]]]],[[[[3],[0],[[[0],[0]]]],[[-,4],[13],[[[-,14],[2]]]],[[-,5],[23],[[[-,28],[4]]]]]]]])

Passo 5

([[[[0],[3],[[[0],[0]]]],[[0],[0],[[[0],[3]]]],[[[[3],[0],[[[0],[0]]]],[[-,4],[13],[[[-,14],[2]]]],[[-,5],[23],[[[-,28],[4]]]]]]]]) underset(L 3 -> - (1)/(14) L 3)(->) ([[[[0],[3],[[[0],[0]]]],[[0],[0],[[[0],[3]]]],[[[[3],[0],[[[0],[0]]]],[[-,4],[13],[[[1],[2]]]],[[-,5],[23],[[[2],[4]]]]]]]])

([[[[0],[3],[[[0],[0]]]],[[0],[0],[[[0],[3]]]],[[[[3],[0],[[[0],[0]]]],[[-,4],[13],[[[1],[2]]]],[[-,5],[23],[[[2],[4]]]]]]]]) underset([[L 1 -> L 1 + 4 L 3],[L 2 -> L 2 - 13 L 3],[L 4 -> L 4 - 2 L 3]])(->) ([[[[0],[3],[[[0],[0]]]],[[0],[0],[[[0],[3]]]],[[[[3],[0],[[[0],[0]]]],[[0],[0],[[[1],[0]]]],[[3],[-,3],[[[2],[0]]]]]]]])

Passo 6

Agora, vamos transformar a matriz ([[[[0],[3],[[[0],[0]]]],[[0],[0],[[[0],[3]]]],[[[[3],[0],[[[0],[0]]]],[[0],[0],[[[1],[0]]]],[[3],[-,3],[[[2],[0]]]]]]]]) num sistema linear e resolvê-lo:

[[3,I_1,=,-,3],[2,I_2,=,0],[[[3,I_3,=,3],[I_4,=,2]]]] -> [[I_1,=,-,1],[I_2,=,0],[[[I_3,=,1],[I_4,=,2]]]]

Resposta

A solução do sistema é (I_1 , I_2 , I_3 , I_4) = (- 1 , 0 , 1 , 2) .

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