Question

(2x-2y - 4z = 2
Resolva o sistema {-x + y + z = 2 por meio do escalonamento de Gauss e marque a
[x-2y+z=-2
alternativa que mostra os valores de x, y e z.
Escolha uma:
a. (-5,2,1)
b.(-1,23)
d.(.6-1)
e. (-5-2,3)​

Answer 1

$\left\{\begin{array}{ccccc}2x&-2y&-4z&=&2\\-x&y&z&=&2\\x&-2y&z&=&-2\end{array}\right\\\\\\L1\leftrightarrow L3\\\\\\\left\{\begin{array}{ccccc}x&-2y&z&=&-2\\-x&y&z&=&2\\2x&-2y&-4z&=&2\end{array}\right\\\\\\L2\leftarrow L2+L1\\\\L3\leftarrow L3-2L1\\\\\\\left\{\begin{array}{ccccc}x&-2y&z&=&-2\\&-y&2z&=&0\\&2y&-6z&=&6\end{array}\right$

$\left\{\begin{array}{ccccc}x&-2y&z&=&-2\\&-y&2z&=&0\\&2y&-6z&=&6\end{array}\right\\\\\\L3\leftarrow L3+2L2\\\\\\\left\{\begin{array}{ccccc}x&-2y&z&=&-2\\&-y&2z&=&0\\&&-2z&=&6\end{array}\right$

Com o sistema escalonado, podemos começar a determinar o valor de cada variável.

$Na~Linha~3:\\\\-2z~=~6\\\\\\z~=~\frac{6}{-2}\\\\\\\boxed{z~=~-3}\\\\\\Na~Linha~2:\\\\-y+2z~=~0\\\\\\-y\,+\,2\,.\,(-3)~=~0\\\\\\-y-6~=~0\\\\\\\boxed{y~=~-6}\\\\\\Na~Linha~1:\\\\x-2y+z~=~-2\\\\\\x\,-\,2\,.\,(-6)\,+\,(-3)~=~-2\\\\\\x+12-3~=~-2\\\\\\\boxed{x~=~-11}$

Solução: (x,y,z) = (-11 , -6 , -3)