Question

Resolva o sistema a seguir, ultilizando a regra de Cramer
x+2y-z=2
2x-y+3z=9
3x+3y-2z=3

Answer 1

$\begin{cases}x+2y-z=2\\2x-y+3z=9\\3x+3y-2z=3\end{cases}$

$\text{D}=\left|\begin{array}{ccc}1&2&-1\\2&-1&3\\3&3&-2\end{array}\right|~\begin{array}{cc}1&~~2\\2&-1\\3&~~3\end{array}$

$\text{D}=1\cdot(-1)\cdot(-2)+2\cdot3\cdot3+(-1)\cdot2\cdot3-(-1)\cdot(-1)\cdot3-1\cdot3\cdot3-2\cdot2\cdot(-2)$
$\text{D}=2+18-6-3-9+8$
$\text{D}=10$

$\text{D}_x=\left|\begin{array}{ccc}2&~~2&-1\\9&-1&~~3\\3&~~3&-2\end{array}\right|~\begin{array}{cc}2&~~2\\9&-1\\3&~~3\end{array}$

$\text{D}_x=2\cdot(-1)\cdot(-2)+2\cdot3\cdot3+(-1)\cdot9\cdot3-(-1)\cdot(-1)\cdot3-2\cdot3\cdot3-2\cdot9\cdot(-2)$
$\text{D}_x=4+18-27-3-18+36$
$\text{D}_x=10$


$\text{D}_y=\left|\begin{array}{ccc}1&2&-1\\2&9&~~3\\3&3&-2\end{array}\right|~\begin{array}{cc}1&2\\2&9\\3&3\end{array}$

$\text{D}_y=1\cdot9\cdot(-2)+2\cdot3\cdot3+(-1)\cdot2\cdot3-(-1)\cdot9\cdot3-1\cdot3\cdot3-2\cdot2\cdot(-2)$
$\text{D}_y=-18+18-6+27-9+8$
$\text{D}_y=20$


$\text{D}_z=\left|\begin{array}{ccc}1&2&2\\2&-1&9\\3&3&3\end{array}\right|~\begin{array}{cc}1&~~2\\2&-1\\3&~~3\end{array}$

$\text{D}_z=1\cdot(-1)\cdot3+2\cdot9\cdot3+2\cdot2\cdot3-2\cdot(-1)\cdot3-1\cdot9\cdot3-2\cdot2\cdot3$
$\text{D}_z=-3+54+12+6-27-12$
$\text{D}_z=30$

Logo:

$x=\dfrac{\text{D}_x}{\text{D}}~\longrightarrow~x=\dfrac{10}{10}~\longrightarrow~\boxed{x=1}$

$y=\dfrac{\text{D}_y}{\text{D}}~\longrightarrow~y=\dfrac{20}{10}~\longrightarrow~\boxed{y=2}$

$z=\dfrac{\text{D}_z}{\text{D}}~\longrightarrow~z=\dfrac{30}{10}~\longrightarrow~\boxed{z=3}$