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Question Number 87831 by aseer imad last updated on 06/Apr/20

Commented by aseer imad last updated on 06/Apr/20

please solve the problem by row echilon form or reduced row echilon form. TIA

Answered by arcana last updated on 06/Apr/20

$$\left(\begin{array}{ccc}1& 1& a\\ 1& a& 1\\ a& 1& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}1\\ 4\\ b\end{array}\right)$$$$$$$$\left(\begin{array}{ccccc}1& 1& a& \mid & 1\\ 1& a& 1& \mid & 4\\ a& 1& 1& \mid & b\end{array}\right)\Rightarrow \left(\begin{array}{ccccc}1& 1& a& \mid & 1\\ 0& a-1& 1-a& \mid & 3\\ 0& 1-a& 1-a^2& \mid & b-a\end{array}\right)$$$$$$$$\Rightarrow \left(\begin{array}{ccccc}1& 1& a& \mid & 1\\ 0& a-1& 1-a& \mid & 3\\ 0& 0& 2-a-a^2& \mid & b-a+3\end{array}\right)$$$$\mathrm{si}a=1\mathrm{el}\mathrm{sistema}\mathrm{es}\mathrm{inconsistentente}$$$$\mathrm{si}a=-2$$$$\bullet 0x+0y+0z=0=b+2+3$$$$\Rightarrow b=-5$$$$\bullet 0x+(-2-1)y+(1-(-2))z=3$$$$-y+z=1\Rightarrow z=1+y$$$$\bullet x+y-2z=1$$$$x+y-2-2y=x-y-2=1$$$$\Rightarrow x=y+3$$$$\mathrm{El}\mathrm{conjunto}\mathrm{de}\mathrm{soluciones}\mathrm{es}$$$$\{\left(\begin{array}{c}x\\ y\\ z\end{array}\right)\in {\mathbb{R}}^3:z=1+y\wedge x=y+3\}=$$$$\{\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)y+\left(\begin{array}{c}3\\ 0\\ 1\end{array}\right):\mathrm{\forall }y\in \mathbb{R}\}$$$$\mathrm{el}\mathrm{par}(-2,-5)\mathrm{hace}\mathrm{que}\mathrm{el}\mathrm{sistema}\mathrm{tenga}$$$$\mathrm{mas}\mathrm{de}\mathrm{una}\mathrm{solucion}.\mathrm{El}\mathrm{sistema}\mathrm{posee}$$$$\mathrm{unica}\mathrm{solucion}\mathrm{para}a\in \mathbb{R}\mathrm{\setminus }\{-2,1\}$$$$$$

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