Ensino Fundamental ⇒ Sistema Lineares Tópico resolvido

[tiberiotavares]

[tex3]\begin{cases}
z +2y+3z-3t=a\\
2x=5y-4x+12t=b\\
7x +y+8z+5t=c
\end{cases}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\begin{cases}
z +2y+3z-3t=a\\
2x=5y-4x+12t=b\\
7x +y+8z+5t=c
\end{cases}[/tex3]

1) prove que o sistema admite solução se e somente se,37a+13b=9c.ache a solução geral do sistema quando a = 2 e b = 4

[petras]

tiberiotavares,
O enunciado tem erros.
[tex3]\begin{cases}
x +2y+3z-3t=a\\
2x-5y-3z+12t=b\\
7x +y+8z+5t=c
\end{cases}\\
\begin{vmatrix}
1 &2&3&-3&a\\
2&-5&-3&12&b\\
7 &1&8&5&c
\end{vmatrix}\\-2L1+L2: -7L1+L3 \implies \begin{vmatrix}
1 &2&3&-3&a\\
0&-9&-9&18&-2a+b\\
0 &-13&-13&26&-7a+c
\end{vmatrix}\\
-\frac{13}{9}L2+L3\implies \begin{vmatrix}
1 &2&3&-3&a\\
0&-9&-9&18&-2a+b\\
0 &0&0&0&\frac{-37a-13b+9c}{9}
\end{vmatrix}\\
\therefore Ter~solução: \frac{-37a-13b+9c}{9}=0 \implies \boxed{37a+13b = 9c}\color{green}\checkmark [/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\begin{cases}
x +2y+3z-3t=a\\
2x-5y-3z+12t=b\\
7x +y+8z+5t=c
\end{cases}\\
\begin{vmatrix}
1 &2&3&-3&a\\
2&-5&-3&12&b\\
7 &1&8&5&c
\end{vmatrix}\\-2L1+L2: -7L1+L3 \implies \begin{vmatrix}
1 &2&3&-3&a\\
0&-9&-9&18&-2a+b\\
0 &-13&-13&26&-7a+c
\end{vmatrix}\\
-\frac{13}{9}L2+L3\implies \begin{vmatrix}
1 &2&3&-3&a\\
0&-9&-9&18&-2a+b\\
0 &0&0&0&\frac{-37a-13b+9c}{9}
\end{vmatrix}\\
\therefore Ter~solução: \frac{-37a-13b+9c}{9}=0 \implies \boxed{37a+13b = 9c}\color{green}\checkmark [/tex3]

[tex3]\begin{vmatrix}
1 &2&3&-3&2\\
2&-5&-3&12&4\\
7 &1&8&5&c
\end{vmatrix}\\-2L1+L2: -7L1+L3 \implies \begin{vmatrix}
1 &2&3&-3&2\\
0&-9&-9&18&0\\
0 &-13&-13&26&c-14
\end{vmatrix}\\
-\frac{L2}{9}.13+L3: -\frac{L2}{9}\implies \begin{vmatrix}
1 &2&3&-3&2\\
0&1&1&-2&0\\
0 &0&0&0&c-14
\end{vmatrix}\\
c\neq 14\implies SI\\
c=14 \implies \boxed{t = n, z =m, y = -m+2n,x = 2-m-n}\color{green}\checkmark

[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\begin{vmatrix}
1 &2&3&-3&2\\
2&-5&-3&12&4\\
7 &1&8&5&c
\end{vmatrix}\\-2L1+L2: -7L1+L3 \implies \begin{vmatrix}
1 &2&3&-3&2\\
0&-9&-9&18&0\\
0 &-13&-13&26&c-14
\end{vmatrix}\\
-\frac{L2}{9}.13+L3: -\frac{L2}{9}\implies \begin{vmatrix}
1 &2&3&-3&2\\
0&1&1&-2&0\\
0 &0&0&0&c-14
\end{vmatrix}\\
c\neq 14\implies SI\\
c=14 \implies \boxed{t = n, z =m, y = -m+2n,x = 2-m-n}\color{green}\checkmark

[/tex3]