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Stagger, classify and solve linear systems a) {x + y - z + t = 1 {3x - y - 2z + t = 2 {-x - 2y + 3z + 2t = -1

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Answer to a math question Stagger, classify and solve linear systems a) {x + y - z + t = 1 {3x - y - 2z + t = 2 {-x - 2y + 3z + 2t = -1

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O sistema de equações dado é: x + y - z + t = 1 3x - y - 2z + t = 2 -x - 2y + 3z + 2t = -1 Primeiro, vamos escalonar o sistema: x + y - z + t = 1 0x + 4y - 5z + 0t = 1 (subtraia 3 vezes a primeira equação da segunda) 0x + 0y + 0z + 0t = 0 (somar a primeira equação à terceira) Agora, o sistema é: x + y - z + t = 1 4y - 5z = 1 0 = 0 A terceira equação é sempre verdadeira e não fornece nenhuma informação nova, portanto podemos ignorá-la. O sistema é subdeterminado, o que significa que possui infinitas soluções. Para encontrar as soluções, podemos expressar x e y em termos de z e t: Da primeira equação, obtemos: x = 1 - y + z - t Da segunda equação, obtemos: y = (1 + 5z) / 4 Portanto, a solução para o sistema é: x = 1 - (1 + 5z) / 4 + z - ty = (1 + 5z) / 4 onde z e t podem ser quaisquer números reais. Isso representa uma família de soluções dependendo dos valores de z e t.

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