Question

Resolva o sistema linear abaixo

x + 2y – z + w + t = 7
3x – y + z – w – t = -1
x + 3y – z + w + t = 9
-x – y + 2z + w + t = -1
x + y – z + 2w – 2t = -5

Answer 1

    $\begin{cases}x+2y-z+w+t=7\\3x-y+z-w-t=-1\\x+3y-z+w+t=9\\-x-y+2z+w+t=-1\\x+y-z+2w-2t=-5\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=7-2y+z-w-t\\3(7-2y+z-w-t)-y+z-w-t=-1\\(7-2y+z-w-t)+3y-z+w+t=9\\-(7-2y+z-w-t)-y+2z+w+t=-1\\(7-2y+z-w-t)+y-z+2w-2t=-5\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=7-2y+z-w-t\\21-6y+3z-3w-3t-y+z-w-t=-1\\7-2y+z-w-t+3y-z+w+t=9\\-7+2y-z+w+t-y+2z+w+t=-1\\7-2y+z-w-t+y-z+2w-2t=-5\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=7-2y+z-w-t\\-7y+4z-4w-4t=-22\\y=2\\y+z+2w+2t=6\\-y+w-3t=-12\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=7-2\times2+z-w-t\\-7\times2+4z-4w-4t=-22\\y=2\\2+z+2w+2t=6\\-2+w-3t=-12\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=7-4+z-w-t\\-14+4z-4w-4t=-22\\y=2\\2+z+2w+2t=6\\-2+w-3t=-12\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=z-w-t+3\\4z-4w-4t=-8\\y=2\\z+2w+2t=4\\w-3t=-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=z-(3t-10)-t+3\\4z-4(3t-10)-4t=-8\\y=2\\z+2(3t-10)+2t=4\\w=3t-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=z-3t+10-t+3\\4z-12t+40-4t=-8\\y=2\\z+6t-20+2t=4\\w=3t-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=z-4t+13\\4z-16t=-48\\y=2\\z+8t=24\\w=3t-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=(-8t+24)-4t+13\\4(-8t+24)-16t=-48\\y=2\\z=-8t+24\\w=3t-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=-8t+24-4t+13\\-32t+96-16t=-48\\y=2\\z=-8t+24\\w=3t-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=-12t+37\\-48t=-144\\y=2\\z=-8t+24\\w=3t-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=-12\times3+37\\t=3\\y=2\\z=-8\times3+24\\w=3\times3-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=-36+37\\t=3\\y=2\\z=-24+24\\w=9-10\end{cases}\Leftrightarrow$

$\Leftrightarrow\begin{cases}x=1\\t=3\\y=2\\z=0\\w=-1\end{cases}$

Resposta:  $\left(x;y;z;w;t\right)=\left(1;2;0;-1;3\right)$

Podes ver mais exercícios sobre resolução de sistemas lineares em:

• https://brainly.com.br/tarefa/32093707
• https://brainly.com.br/tarefa/32072039
• https://brainly.com.br/tarefa/31726917

Answer 2

x + 2y – z + w + t = 7 ①

3x – y + z – w – t = −1 ②

x + 3y – z + w + t = 9 ③

−x – y + 2z + w + t = −1 ④

x + y – z + 2w – 2t = −5 ⑤

• Observe que somando ou subtraindo algumas equações membro a membro três incógnitas são eliminadas, portanto esse parece ser a forma de resolução mais rápida.

① + ② ⇒ 4x + y = 6 ⑥

② + ③ ⇒ 4x + 2y = 8 ⑦

⑦ − ⑥ ⇒ y = 2

Substitua y em ⑥ ⇒ 4x + y = 6 ⇒ 4x + 2 = 6 ⇒ 4x = 4 ⇒ x = 1

④ − ③ ⇒ −2x − 4y + 3z = −10 ⇒ −2 − 8 + 3z = −10 ⇒ z = 0

⑤ + 2×④ ⇒ −x − y + 3z + 4w = −7 ⇒ −1 − 2 + 4w = −7 ⇒ w = −1

Em ①: x + 2y – z + w + t = 7  ⇒ 1 + 2•2 − 1 + t = 7 ⇒ t  = 3

Solução:

$\Large \begin{cases} \sf x=1 \\ \sf y=2 \\ \sf z=0 \\ \sf t=3 \\ \sf w=-1 \end{cases}$

Veja outros casos em:

https://brainly.com.br/tarefa/36001747

https://brainly.com.br/tarefa/35536409

https://brainly.com.br/tarefa/30181699

https://brainly.com.br/tarefa/30182866