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»SISTEME DE ECUATII-gimnaziu

Data publicarii: 21 Februarie, 2012

EXERCITIUL 3

Suport teoretic:

Sisteme liniare, regula lui Cramer.

Enunt:

Sa se rezolve urmatorul sistem liniar folosind regula lui Cramer:

$\begin{cases}4x-3y=11\\x+5y=-3\end{cases}.$ \begin{cases}4x-3y=11\\x+5y=-3\end{cases}.

Raspuns:

S = {(2;-1)}

Rezolvare:

$\Delta=\begin{vmatrix}4&-3\\1&5\end{vmatrix}=4\cdot5-(-3)\cdot1=20+3=23\not={0}.$ \Delta=\begin{vmatrix}4&-3\\1&5\end{vmatrix}=4\cdot5-(-3)\cdot1=20+3=23\not={0}.

$\Delta_x=\begin{vmatrix}11&-3\\-3&5\end{vmatrix}=11\cdot5-(-3)\cdot(-3)=55-9=46$ \Delta_x=\begin{vmatrix}11&-3\\-3&5\end{vmatrix}=11\cdot5-(-3)\cdot(-3)=55-9=46

$\Delta_y=\begin{vmatrix}4&11\\1&-3\end{vmatrix}=4\cdot(-3)-11\cdot1=-12-11=-23.$ \Delta_y=\begin{vmatrix}4&11\\1&-3\end{vmatrix}=4\cdot(-3)-11\cdot1=-12-11=-23.

Deci sistemul are solutia unica data de formulele:

$x=\frac{\Delta_x}{\Delta}=\frac{46}{23}=2\;si\;y=\frac{\Delta_y}{\Delta}=\frac{-23}{23}=-1.$ x=\frac{\Delta_x}{\Delta}=\frac{46}{23}=2\;si\;y=\frac{\Delta_y}{\Delta}=\frac{-23}{23}=-1.