12. foglalkozás: Háromismeretlenes lineáris egyenletrendszer

\begin{equation}
\left.
\begin{array}{crcl}
(1) & a_{11}x_1 + a_{12}x_2+a_{13}x_3 &=& b_1\\
(2) & a_{21}x_1 + a_{22}x_2+a_{23}x_3 &=& b_2\\
(3) & a_{31}x_1 + a_{32}x_2+a_{33}x_3 &=& b_3\\
\hline
\end{array}
\right\}
\end{equation}
Az (1) és a (2) egyenletpárból ejtsük ki \(x_3\)-at:
\begin{equation}
\left.
\begin{array}{crcll}
(1) & a_{11}x_1 + a_{12}x_2+a_{13}x_3 &=& b_1 & \big/\,\cdot a_{23}\\
(2) & a_{21}x_1 + a_{22}x_2+a_{23}x_3 &=& b_2 & \big/\,\cdot a_{13}\\
\hline
\end{array}
\right\}\\\\
\left.
\begin{array}{crcll}
(1') & a_{11}a_{23}x_1 + a_{12}a_{23}x_2+a_{13}a_{23}x_3 &=& b_1a_{23} & \\
(2') & a_{13}a_{21}x_1 + a_{13}a_{22}x_2+a_{13}a_{23}x_3 &=& a_{13}b_2 & \\
\hline
\end{array}
\right\}\\\\
\begin{array}{crcl}
(1^*):(1')-(2') & (a_{11}a_{23}-a_{13}a_{21})x_1 + (a_{12}a_{23}-a_{13}a_{22})x_2 &=& b_1a_{23}-a_{13}b_2
\end{array}
\end{equation}
Az (1) és a (3) egyenletbõl is ejtsük ki \(x_3\)-at:
\begin{equation}
\left.
\begin{array}{crcll}
(1) & a_{11}x_1 + a_{12}x_2+a_{13}x_3 &=& b_1 & \big/\,\cdot a_{33}\\
(3) & a_{31}x_1 + a_{32}x_2+a_{33}x_3 &=& b_3 & \big/\,\cdot a_{13}\\
\hline
\end{array}
\right\}\\\\
\left.
\begin{array}{crcll}
(1') & a_{11}a_{33}x_1 + a_{12}a_{33}x_2+a_{13}a_{33}x_3 &=& b_1a_{33} & \\
(3') & a_{13}a_{31}x_1 + a_{13}a_{32}x_2+a_{13}a_{33}x_3 &=& a_{13}b_3 & \\
\hline
\end{array}
\right\}\\\\
\begin{array}{crcll}
(2^*):(1)-(3) & (a_{11}a_{33}-a_{13}a_{31})x_1 + (a_{12}a_{33}-a_{13}a_{32})x_2&=& b_1a_{33}-a_{13}b_3 &
\end{array}
\end{equation}
A keletkezett (1*) és (2*) egyenletpár egy kétismeretlenes egyenletrendszer:
\begin{equation}
\left.
\begin{array}{crcl}
(1^*) & (a_{11}a_{23}-a_{13}a_{21})x_1 + (a_{12}a_{23}-a_{13}a_{22})x_2 &=& b_1a_{23}-a_{13}b_2\\
(2^*) & (a_{11}a_{33}-a_{13}a_{31})x_1 + (a_{12}a_{33}-a_{13}a_{32})x_2&=& b_1a_{33}-a_{13}b_3\\
\hline
\end{array}
\right\}
\end{equation}
Ejtsük ki \(x_2\)-t:
\begin{equation}
\left.
\begin{array}{crcll}
(1^*) & (a_{11}a_{23}-a_{13}a_{21})x_1 + (a_{12}a_{23}-a_{13}a_{22})x_2 &=& b_1a_{23}-a_{13}b_2 & \big/\,\cdot(a_{12}a_{33}-a_{13}a_{32})\\
(2^*) & (a_{11}a_{33}-a_{13}a_{31})x_1 + (a_{12}a_{33}-a_{13}a_{32})x_2&=& b_1a_{33}-a_{13}b_3 & \big/\,\cdot(a_{12}a_{23}-a_{13}a_{22})\\
\hline
\end{array}
\right\}\\\\
\left.
\begin{array}{crcl}
(1^*) & (a_{11}a_{23}-a_{13}a_{21})(a_{12}a_{33}-a_{13}a_{32})x_1 + (a_{12}a_{23}-a_{13}a_{22})(a_{12}a_{33}-a_{13}a_{32})x_2 &=& (b_1a_{23}-a_{13}b_2)(a_{12}a_{33}-a_{13}a_{32})\\
(2^*) & (a_{11}a_{33}-a_{13}a_{31})(a_{12}a_{23}-a_{13}a_{22})x_1 + (a_{12}a_{33}-a_{13}a_{32})(a_{12}a_{23}-a_{13}a_{22})x_2&=& (b_1a_{33}-a_{13}b_3)(a_{12}a_{23}-a_{13}a_{22})\\
\hline
\end{array}
\right\}\\\\
\begin{array}{rcl}
\big[(a_{11}a_{23}-a_{13}a_{21})(a_{12}a_{33}-a_{13}a_{32})-(a_{11}a_{33}-a_{13}a_{31})(a_{12}a_{23}-a_{13}a_{22})\big]x_1 &=& (b_1a_{23}-a_{13}b_2)(a_{12}a_{33}-a_{13}a_{32})- (b_1a_{33}-a_{13}b_3)(a_{12}a_{23}-a_{13}a_{22})
\end{array}
\end{equation}

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