Resolva o sistema abaixo

[[[0,,,,,1,,,,2,,,,,1,,,,,1,|,,,,,,,6],[0,-,2,-,4,-,2,-,4,|,-,18],[0,,,,1,,,,2,,,,2,,,,3,|,,,,13]]]

Passo 1

Inicialmente vamos escalonar a matriz:

[[[0,,,,,1,,,,2,,,,,1,,,,,1,|,,,,,,,6],[0,-,2,-,4,-,2,-,4,|,-,18],[0,,,,1,,,,2,,,,2,,,,3,|,,,,13]]] overset(l_2 = l_2 + 2 l_1)(->) [[[0,,,,,1,,,,2,,,,,1,,,,,1,|,,,,,,,6],[0,,,,0,,,,0,,,,0,-,2,|,,,-,6],[0,,,,1,,,,2,,,,2,,,,3,|,,,,13]]] overset(l_3 = l_3 - l_1)(->)

[[[0,,,,,1,,,,2,,,,,1,,,,,1,|,,,,,,,6],[0,,,,0,,,,0,,,,0,-,2,|,,,-,6],[0,,,,0,,,,0,,,,1,,,,2,|,,,,,,7]]] overset(l_2 harr l_3)(->) [[[0,,,,,1,,,,2,,,,1,,,,,1,|,,,,,,,6],[0,,,,0,,,,0,,,,1,,,,,2,|,,,,,,,7],[0,,,,0,,,,0,,,,0,,-,2,|,,,,,-,6]]] overset(l_1 = l_1 - l_2)(->)

[[[0,,,,,1,,,,2,,,,,0,,,,-,1,|,-,1],[0,,,,0,,,,0,,,,1,,,,,2,|,,,,7],[0,,,,0,,,,0,,,,0,,-,2,|,-,6]]] overset(l_3 = (- (1)/(2)) l_3)(->) [[[0,,,,,1,,,,2,,,,,0,,,,-,1,|,-,1],[0,,,,0,,,,0,,,,1,,,,,2,|,,,,7],[0,,,,0,,,,0,,,,0,,,,,,1,|,,,,3]]] overset(l_1 = l_1 + l_3)(->)

[[[0,,,,,1,,,,2,,,,,0,,,,,,0,|,,,,2],[0,,,,0,,,,0,,,,1,,,,,2,|,,,,7],[0,,,,0,,,,0,,,,0,,,,,1,|,,,,3]]] overset(l_2 = l_2 - 2 l_1)(->) [[[0,,,,,1,,,,2,,,,,0,,,,,,0,|,,,,2],[0,,,,0,,,,0,,,,1,,,,,0,|,,,,1],[0,,,,0,,,,0,,,,0,,,,,1,|,,,,3]]]

Passo 2

Sabemos que o sistema relativo à matriz do enunciado é equivalente ao sistema relativo à matriz escalonada, então vamos resolver esse último sistema.

Sejam x_1 , x_2 , x_3 , x_4 e x_5 as variáveis do sistema. Como os pivôs da matriz aparecem na segunda, quarta e quinta coluna segue que x_2 , x_4 e x_5 são variáveis dependentes, enquanto x_1 e x_3 são variáveis livres. Definimos x_1 = s e x_3 = t . Agora, vamos descobrir o valor das variáveis dependentes. Pela terceira equação do sistema temos que x_5 = 3 , pela segunda temos que x_4 = 1 e pela primeira temos x_2 + 2 x_3 = 2 => x_2 + 2 t = 2 => x_2 = 2 - 2 t . Logo, a solução do sistema é (s , 2 - 2 t , t , 1 , 3) = (0 , 2 , 0 , 1 , 3) + s (1 , 0 , 0 , 0 , 0) + t (0 , - 2 , 1 , , 0 , 0) = (0 , 2 , 0 , 1 , 3) + < (1 , 0 , 0 , 0 , 0) , (0 , - 2 , 1 , 0 , 0) > .

Resposta

(0 , 2 , 0 , 1 , 3) + < (1 , 0 , 0 , 0 , 0) , (0 , - 2 , 1 , 0 , 0) >