Question

Pentru m=2 Sa se rezolve sistemul;
{ x+y+z=0
x+my+z=0
x+y+mz=0

Answer 1

$\displaysyle \mathtt{ \left\{\begin{array}{ccc}\mathtt{x+y+z=0}\\\mathtt{x+my+z=0}\\\mathtt{x+y+mz=0}\end{array}\right}\\ \\ \mathtt{m=2 \Rightarrow \left\{\begin{array}{ccc}\mathtt{x+y+z=0}\\\mathtt{x+2y+z=0}\\\mathtt{x+y+2z=0}\end{array}\right \Rightarrow A= \left(\begin{array}{ccc}1&1&1\\1&2&1\\1&1&2\end{array}\right)}$

$\displaystyle \mathtt{\Delta=\left|\begin{array}{ccc}1&1&1\\1&2&1\\1&1&2\end{array}\right|=1 \cdot 2 \cdot 2+1 \cdot 1 \cdot 1+1 \cdot 1 \cdot 1-1 \cdot 2 \cdot 1-1 \cdot 1 \cdot 2-}\\ \\ \mathtt{~~~~~~~~~~~~~~~~~~~~~~~~~~~~-1 \cdot 1 \cdot 1=4+1+1-2-2-1=1}\\ \\ \mathtt{\Delta=1 \not = 0}$

$\displaystyle \mathtt{\Delta_x=\left|\begin{array}{ccc}0&1&1\\0&2&1\\0&1&2\end{array}\right|=0 \cdot 2 \cdot 2+1 \cdot 0 \cdot 1+1 \cdot 1 \cdot 0-1 \cdot 2 \cdot 0-1 \cdot 0 \cdot 2-}\\ \\ \mathtt{~~~~~~~~~~~~~~~~~~~~~~~~~~~~-0 \cdot 1 \cdot 1=0}\\ \\ \mathtt{\Delta_x=0}$

$\displaystyle \mathtt{\Delta_y=\left|\begin{array}{ccc}1&0&1\\1&0&1\\1&0&2\end{array}\right|=1 \cdot 0 \cdot 2+1 \cdot 1 \cdot 0+0 \cdot 1 \cdot 1-1 \cdot 0 \cdot 1-0 \cdot 1 \cdot 2-}\\ \\ \mathtt{~~~~~~~~~~~~~~~~~~~~~~~~~~~~-1 \cdot 1 \cdot 0=0}\\ \\ \mathtt{\Delta_y=0}$

$\displaystyle \mathtt{\Delta_z= \left|\begin{array}{ccc}1&1&0\\1&2&0\\1&1&0\end{array}\right|=1 \cdot 2 \cdot 0+0 \cdot 1 \cdot 1+1 \cdot 0 \cdot 1-0 \cdot 2 \cdot 1-1 \cdot 1 \cdot 0-}\\ \\ \mathtt{~~~~~~~~~~~~~~~~~~~~~~~~~~~~-1 \cdot 0 \cdot 1=0}\\ \\ \mathtt{\Delta_z=0}\\ \\ \mathtt{x= \frac{\Delta_x}{\Delta} = \frac{0}{1}=0 }\\ \\ \mathtt{y= \frac{\Delta_y}{\Delta}= \frac{0}{1}=0 }\\ \\ \mathtt{z= \frac{\Delta_z}{\Delta}= \frac{0}{1}=0}\\ \\ \mathtt{x=0;~y=0;~z=0}$