How many minutes does Train $1$ take for the entire trip? 
Two trains simultaneously depart towards each other on one route of Washington's Metro line. Train $1$ departs from Point A on one end and Train $2$ departs from Point B on the other end. At the instant the trains pass each other, it takes Train $1$ another $16$ minutes to reach the point B and it takes Train $2$ another $9$ minutes to reach Point A. How many minutes does Train $1$ take for the entire trip?

Let the speed of Train $1$ be $s_1$ and let the speed of Train $2$ be $s_2$. Then we have $$s_1t+16s_1 = s_2t+9s_2,$$ which gives $t(s_1-s_2) = 9s_2-16s_1$ and so $t = \dfrac{9s_2-16s_1}{s_1-s_2}$. Thus, $t+16 = -\dfrac{5s_2}{s_1-s_2}$.
It doesn't seem possible to get the answer from here, but the answer is $28$. What did I do wrong?

So you let the speed of train $1$ be $s_1$ and train $2$ be $s_2$. Let us call the distance between $A$ and $B$ as $x$. Let the distance between their common meeting point $C$ and $A$ be $y$, so that $BC$ is then $x-y$.
Then,
1) It is clear that train $2$ takes $7$ minutes  less than train $1$. So one equation would be $\frac{x}{s_1} = \frac{x}{s_2} + 7$.
2) The time taken for train $2$ to travel $x-y$ is the same as time taken for train $1$ to travel $y$. So then $\frac{y}{s_1} = \frac{x-y}{s_2}$, or that $ys_2=(x-y)s_1$ or that $y(s_2+s_1) = x$.
3)Train $2$ travels distance $y$ in nine minutes, so $\frac{y}{s_2} = 9$.
4)Train $1$ travels distance $x-y$ in sixteen minutes, so $\frac{x-y}{s_1} = 16$.
From the third point, $y = 9s_2$, and from the fourth, $x-y = 16s_1$, so that $x=16s_1+9s_2$.
Also, from the second point, $9s_2(s_2+s_1) = x$.
From $x=16s_1+9s_2$, we insert this in the first equation to see that $16+\frac{9s_2}{s_1} = 9+\frac{16s_1}{s_2} + 7$, so that $16s_1^2 = 9s_2^2$. Hence, $\frac{s_1}{s_2} = \frac 34$.
Now, substitute for $s_2$ in $x = 16s_1 + 9s_2$. We know that $3s_2 = 4s_1$ ,so that $9s_2 = 12s_1$. Hence, $x = 16s_1 + 12s_1 = 28s_1$. Hence, $\frac{x}{s_1} = 28$.
This means that train $1$ travels the entire distance in $28$ minutes. Train $2$ would do it in $21$ minutes.
