The equations 2 x+3 y+4=0 ; 3 x+4 y+6=0 and 4 x+5 y+8=0 are

Consider first two equations: $2 x+3 y=-4 \quad$ and $\quad 3 x+4 y=-6$ We have \Delta=\left|$\begin{array}{ll}2 & 3 \\ 3 & 4\end{array}$\right|=-1 \neq 0 \Delta_{\mathrm{x}}=\left|$\begin{array}{ll} -4 & 3 \\ -6 & 4 \end{array}$\right|=2 \quad\text{ and } \quad \Delta_{\mathrm{y}}=\left|$$\begin{array}{cc}2 & -4 \\ 3 & -6 \end{array}$$\right|=0 $\therefore x=-2$ and $y=0$ Now this solution satisfies the third, so the equations are consistent with unique solution.