Q.Using determinants, examine whether the system $x-2y=4,\ 2x-4y=5$ is consistent. If it is not, explain why.

[!TLDR] Compute the coefficient determinant $D$ and the constant-replaced determinant $D_1$. [!ANSWER] $D=0$ but $D_1=- …

For $x-2y=4,\ 2x-4y=5$: $D=\begin{vmatrix}1&-2\2&-4\end{vmatrix}=1(-4)-(-2)(2)=-4+4=0$. Since $D=0$, there is no unique solution; checking $D_1=\begin{vmatrix}4&-2\5&-4\end{vmatrix}=4(-4)-(-2)(5)=-16+10=-6 e0$. Since $D=0$ but $D_1 e0$, the system is inconsistent: it has no solution. Geometrically, the two equations describe parallel (but distinct) lines, since the second equation is …

Compute the coefficient determinant $D$ first; if $D=0$, compute $D_1$ (and $D_2$ if needed) to decide between "inconsistent" ($D_1$ or $D …

Stopping after finding $D=0$ and concluding "infinitely many solutions" without checking $D_1$ -- $D=0$ alone only rules out a unique solution, it d …