Q:

I can't understand result in my textbook.

I've the following equation in my physics text book:
given that 
$$t=t_1+t_2+t_3,$$$$t_2=t_3$$ $$s_1 = v_1t_1,$$ $$s_2 =v_2t_2;$$ $$s_3 = v_3t_3,$$$$s_2+s_3=\frac{s}{2}=s_1$$
$$t= t_1 + 2t_2 \implies t_2=\frac{t-t_1}{2}.$$ From here starts the part that I don't understand:
$$s_1=\frac{s}{2}=s_2+s_3= v_2t_2+v_3t_3=(v_2+v_3)\frac{t-t_1}{2}=v_1t_1$$
from here they jump to:
$$t=\frac{(2v_1+v_2+v_3)t_1}{v_2+v_3}$$
I don't understand how they made this transition, I'm skipping some little detail and I can't find it.Any hits which will make it easier ?

A:

Take the last two parts to the chain equality and divide both sides by $(v_2+v_3)/2$ to get
$$t-t_1=\frac{2v_1t_1}{v_2+v_3}$$ then adding $t_1$ to both sides gives $$t=\left(\frac{2v_1}{v_2+v_3}+1\right)t_1$$
Now just simplify the term in parens by changing the $1$ to $\frac{v_2+v_3}{v_2+v_3}$

