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Question Number 81876 by zainal tanjung last updated on 16/Feb/20

$$\mathrm{The}\mathrm{system}\mathrm{of}\mathrm{linear}\mathrm{equations}x+y+z=2,$$$$2x+y-z=3,3x+2y+kz=4\mathrm{has}\mathrm{a}\mathrm{unique}$$$$\mathrm{solution}\mathrm{if}$$

Commented by Tony Lin last updated on 16/Feb/20

$$\mathrm{\Delta }=\left|\begin{array}{ccc}1& 1& 1\\ 2& 1& -1\\ 3& 2& k\end{array}\right|$$$$=k-3+4-3+2-2k$$$$=-k\ne 0$$$$\Rightarrow k\ne 0$$$${\mathrm{\Delta }}_x=\left|\begin{array}{ccc}2& 1& 1\\ 3& 1& -1\\ 4& 2& 0\end{array}\right|$$$$=6-4-4+4=2\ne 0$$$$\Rightarrow whenk=0,theequationhasnosolution$$$$k\ne 0,theequationhasauniquesolution$$

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