Question

Escalone e resolva o sistema
x+y=-1
x+z=1
y+z=4​

Answer 1

Vamos começar organizando as equações:

$\left\{\begin{array}{ccccc}x&+~~~y&&=&-1\\x&&+~~~z&=&1\\&~~~~~y&+~~~z&=&4\end{array}\right$

Agora, montando a matriz dos coeficientes:

$\left[\begin{array}{ccccc}1&1&0&|&-1\\1&0&1&|&1\\0&1&1&|&4\end{array}\right]$

Agora sim, começando o escalonamento:

$L2\leftarrow L2-L1\\\\\\\left[\begin{array}{ccccc}1&1&0&|&-1\\0&-1&1&|&2\\0&1&1&|&4\end{array}\right]\\\\\\L3\leftarrow L3+L2\\\\\\\left[\begin{array}{ccccc}1&1&0&|&-1\\0&-1&1&|&2\\0&0&2&|&6\end{array}\right]$

Com a matriz já escalonada, podemos remontar o sistema:

$\left\{\begin{array}{ccccc}x&+~~~y&&=&-1\\&-~~~y&+~~~z&=&2\\&&+~~2z&=&6\end{array}\right\\\\\\Pela~3^a~equacao,~temos:\\\\2z~=~6\\\\\\z~=~\frac{6}{2}\\\\\\\boxed{z~=~3}\\\\\\Pela~2^a~equacao,~temos:\\\\-y~+~z~=~2\\\\\\-y~+~3~=~2\\\\\\-y~=~-1\\\\\\\boxed{y~=~1}\\\\\\Pela~1^a~equacao,~temos:\\\\x~+~y~=~-1\\\\\\x~+~1~=~-1\\\\\\\boxed{x~=\,-2}$

Resposta: (x,y,z) = (-2 , 1 , 3)