Question

$\left \{ \begin{array}{c}2x-3y+z=1\\3x-3y-6z=0\\7x-2y-9z=2\end{array}$ Qual o conjunto solução?

Answer 1

utilizando a regra de crammer
$\left[\begin{array}{ccc}2&-3&1\\3&-3&-6\\7&-2&-9\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}1\\0\\2\end{array}\right]$

 
temos

$A= \left[\begin{array}{ccc}2&-3&1\\3&-3&-6\\7&-2&-9\end{array}\right] \\\\\\\boxed{det(A)=90}$


resolvendo

$A_x= \left[\begin{array}{ccc}\bf1&-3&1\\\bf0&-3&-6\\ \bf2&-2&-9\end{array}\right] \to det(A_x)= 57\\\\\\\\A_y= \left[\begin{array}{ccc}2&\bf1&1\\3&\bf0&-6\\7&\bf2&-9\end{array}\right] \to det(A_y)=15\\\\\\\\A_z= \left[\begin{array}{ccc}2&-3&\bf1\\3&-3&\bf0\\7&-2&\bf2\end{array}\right] \to det(A_z)=21$

a solução procurada será

$x= \frac{det(A_x)}{det(A)}= \frac{57}{90}= \frac{19}{30} \\\\\\y= \frac{det(A_y)}{det(A)}= \frac{15}{90} = \frac{1}{6} \\\\\\z= \frac{det(A_z)}{det(A)}= \frac{21}{90}= \frac{7}{30}$