The set of equations $2 x-3 y=4,7 x-2 y=2,9 x-5 y=[4 a]$ may be consistent, then [a] is (where [.] d

Question

The set of equations $2 x-3 y=4,7 x-2 y=2,9 x-5 y=[4 a]$ may be consistent, then [a] is (where [.] denotes G.I.F)

Professor PC Thomas Classes · Answer

$\left|\begin{array}{ccc}2 & -3 & -4 \ 7 & -2 & -2 \ 9 & -5 & -[4 a]\end{array}ight|=0$ Applying $C_3 ightarrow C_3-C_2$, then $ \begin{aligned} & \left|\begin{array}{ccc} 2 & -3 & -1 \ 7 & -2 & 0 \ 9 & -5 & -[4 a]+5 \end{array}ight|=0 \ & \Rightarrow-1(-17)+(-[4 a]+5)(17)=0 \ & \Rightarrow 17+17(-[4 a]+5)=0 \ & 	herefore[4 a]=6 \ & 	ext { or } 6 \leq 4 a<7 \ & \Rightarrow \frac{3}{2} \leq a<\frac{7}{4} \ & 	herefore a \in\left[\frac{3}{2}, \frac{7}{4}ight) \end{aligned} $