Question 26774
The average speed of each train is the distance divided by the elapsed time 
for each.
The first thing we can write down is 
{{{v[a] = v[c] + 50}}}
How long is each train moving at their average speeds?
The commuter train runs from noon to 3PM, so
{{{t[c] = 3}}}
The Amtrak runs from noon + 2 hours to 3 PM, or 2 PM to 3 PM, so
{{{t[a] = 1}}}
What else happens at 3 PM?
The Amtrak train is 10 miles behind the commuter train, so
{{{d[a] = d[c] - 10}}}
Now come the vital equations:
(a) {{{d[a] = v[a] * t[a]}}}
and
(b) {{{d[c] = v[c] * t[c]}}}
Substitute {{{d[a] = d[c] - 10}}} for d[a]
Substitute {{{v[a] = v[c] + 50}}} for v[a]
Substitute {{{t[a] = 1}}} for t[a]
so,
{{{d[c] - 10 = (v[c] + 50) * 1}}}
And for equation (b),
Substitute {{{t[c] = 3}}}
{{{d[c] = v[c] * 3}}}
So, now,
{{{(v[c] * 3) - 10 = (v[c] + 50) * 1}}}
Now you can solve for v[c]
{{{2 * v[c] = 60}}}
{{{v[c] = 30}}}
Since {{{v[a] = v[c] + 50}}}, v[a] = 80
This answers how fast each train is going
The Commuter is going 30 MPH
The Amtrak is going 80 MPH
How far from Chicago are the trains?
{{{d[a] = v[a] * t[a]}}}
{{{d[a] = 80 * 1}}}
d[a] = 80 miles
{{{d[c] = v[c] * t[c]}}}
{{{d[c] = 30 * 3}}}
d[c] = 90 miles