The sum of the two trains is 723.5 mph if the speed of the first train is 12.5 mph faster than that of the second train find the speeds of each

First, let's see if we can rewrite this word problem a little bit more mathematically. We won't get to mathy or technical so no worries. We just want to look at it in a more straightforward way, if we can.

Train A's mph plus Train B's mph summed equal 723.5 mph. Train A's mph is greater than Train B's mph by 12.5 mph.   
So what should we do to solve this problem? Since we are dealing with two of something and we know the value of the two combined, it might make sense to start by dividing that value by 2.
723.5 / 2 =  361.75. If the two trains were travelling at the same speed, we would be done. Unfortunately, they are not so we need to think about this some more. 
Train A is going 12.5 mph faster than Train B. Let's rewrite.
Train A mph = 12.5 + 361.75 = 374.25  Okay, so Train A is travelling at a speed of 374.25 mph. So we're done right? Not exactly. We are asked to fing the speeds of BOTH trains. How do we find the speed of Train B? We have added a portion of the combined total to Train A. It seems to follow, then, we should probably subtract the same portion from Train A. What are we going to do? You guessed it! Rewrite.
Train B mph = 361.75 - 12.5 = 349.25 HA HA! We seem to have figured it out. Let's do one last thing to check our work. Let's add the two trains' speeds together. If we did this right, we should get our summed value of 723.5 mph
374.25 + 349.25 = 723.5
Pat yourself on the back! We did it!

374.25 + 349.25 =

X is speed of first train
y is speed of 2nd train

let's say x>y
oh way, first is 12.5speed faster than 2nd
x=12.5+y

alrighty
sum is 723.5
x+y=723.5
and x=12.5+y

subsitute 12.5+y for x in other eqaution

12.5+y+y=723.5
12.5+2y=723.5
minus 12.5 from both sides
2y=711
divide both sides by 2
y=355.5

sub back
x=12.5+y
x=12.5+355.5
x=368

the first train's speed is 368 mph
the 2nd train's speed is 355.5 mph