The point of intersection of the lines _{$\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$} and _{$\frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}$} is

Let _{$\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}=\lambda$} so that a general point on it is _{$P\left(3\lambda -1,5\lambda -3,7\lambda -5\right)$} let _{$\frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}=\mu$}

So that general point on it is_{$Q\left(\mu +2,3\mu +4,5\mu +6\right)$} . The two lines intersect P and Q coincide _{$\therefore 3\lambda -1=\mu +2,5\lambda -3=3\mu +4,7\lambda -5=5\mu +6$}

Solving first two equations, we get _{$\lambda =\frac{1}{2},\mu =-\frac{3}{2}$}

These values satisfy the third equation. hence, coordinates of point  of intersection are _{$\left(\frac{1}{2},-\frac{1}{2},-\frac{3}{2}\right)$}