Find the value of p, if the following lines are concurrent
$3\mathrm{x}+4\mathrm{y}=5,2\mathrm{x}+3\mathrm{y}=4,\mathrm{px}+4\mathrm{y}=6$

Given lines are  $3\mathrm{x}+4\mathrm{y}=5 ...\left(1\right); 2\mathrm{x}+3\mathrm{y}=4 ...\left(2\right)$
$\mathrm{px}+4\mathrm{y}=6 ....\left(3\right)$
Let p(x, y) be the point of concurrence
solving (1) and (2)
$\begin{array}{cccc} \mathrm{x}& \mathrm{y}& 1 & \\ 4& -5& 3& 4\\ 3& -4& 2& 3\end{array}$
$$\begin{gathered} \frac{\mathrm{x}}{-16+15}=\frac{\mathrm{y}}{-10+12}=\frac{1}{9-8} \\ \Rightarrow \mathrm{x}=-1,\mathrm{y}=2 \end{gathered}$$
the point of concurrence is (–1, 2)
From equation (3), $$\begin{gathered} \mathrm{p}(-1)+4\left(2\right)=6 \\ \Rightarrow \mathrm{p}=2 \end{gathered}$$