For what value of $a$ are the lines $3x + y + 2 = 0$, $2x - y+ 3 = 0$ and $x + ay - 3 = 0$ are concurrent.
Equations of the given lines are
$3x + y + 2 = 0$ $\;\;\; \cdots \; (1)$; $\;$ $2x - y + 3 = 0$ $\;\;\; \cdots \; (2)$; $\;$ $x + ay - 3 = 0$ $\;\;\; \cdots \; (3)$
Solving equations $(1)$ and $(2)$ simultaneously we get
$5x + 5 = 0$ $\implies$ $x = -1$
and $y = 2 \times \left(-1\right) + 3 = 1$
$\therefore \;$ The point of intersection of equations $(1)$ and $(2)$ is $\left(x, y\right) = \left(-1, 1\right)$
$\because \;$ Equations $(1)$, $(2)$ and $(3)$ are concurrent, the point $\left(x,y\right)$ also satisfies equation $(3)$.
Substituting $\left(x,y\right)$ in equation $(3)$, we get
$-1 + a -3 = 0$ $\implies$ $a = 4$