Q.The system of equations $2x + 3y = 7$ and $4x + 6y = 5$ is:

Consistency is read off from $D$ and $D_x$: here $D=0$ but $D_x\neq0$, which means no solution.

Coefficient determinant: $D = \begin{vmatrix} 2 & 3 \\ 4 & 6 \end{vmatrix} = (2)(6)-(3)(4) = 12-12 = 0.$

Because $D=0$, there is no unique solution — we must check $D_x$ (and $D_y$) to decide between 'no solution' and 'infinitely many'. $D_x = \begin{vmatrix} 7 & 3 \\ 5 & 6 \end{vmatrix} = (7)(6)-(3)(5) = 42-15 = 27 \neq 0.$

Since $D=0$ but $D_x\neq0$, the system is inconsistent — it has no solution.

Checking the options. Option (a) needs $D\neq0$, which fails. Option (b) needs $D=D_x=D_y=0$, but $D_x=27\neq0$, so it fails too. Option (c) is exactly the case $D=0$ with some determinant non-zero, so it is correct; option (d) is therefore ruled out. …

The trap is to see D = 0 and immediately pick 'infinitely many solutions'. D = 0 only rules out a unique solution; whether there are infinitely many (all of D, Dx, Dy zero) or n …