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Let $\alpha, \beta \in R$ be such that the system of linear equations $x+2 y+z=5,2 x+y+\alpha z=5,8 x+4 y+\beta z=18$ has no

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Solution

for no solution $\Delta=\left|\begin{array}{lll}1 & 2 & 1 \\ 2 & 1 & \alpha \\ 8 & 4 & \beta\end{array}\right|=4 \alpha-\beta=0 \Rightarrow \beta=4 \alpha$
Second eq. $2 \mathrm{x}+\mathrm{y}+\alpha \mathrm{z}=5$
third eq. $8 x+4 y+4 \alpha z=18 \Rightarrow 2 x+y+\alpha z=\frac{9}{2}$ both are parallel planes $\therefore$ no solution
[CHAPTER M26 MATRICES TOPIC T2 ]

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