The equation of the line passing through the point of intersection of $2x+3y+4=0,7x+5y=3$  and $(1,4)$  is 

$\begin{array}{l}{L}_1+\lambda L_2=0\\ (2x+3y+4)+\lambda (7x+5y-3)=0\end{array}$

It passing through $(1,4)$

$\begin{array}{l}[2\left(1\right)+3\left(4\right)+4]+\lambda (7\left(1\right)+5\left(4\right)-3)=0\\ (2+12+4)+\lambda (7+20-3)=0\\ \lambda (27-3)=-18\Rightarrow \lambda =\frac{-18}{24}=\frac{-3}{4}\\ (2x+3y+4)-\frac{3}{4}(7x+5y-3)=0\\ 8x+12y+16-21x-15y+9=0\Rightarrow -13x-3y+25=0\\ 13x+3y-25=0\end{array}$