Question

HELP! Two trains started at the same time from stations which were 360 miles apart and traveled toward each other. The rate of the fast train exceeded the rate of the slow train 10 mph. At the end of two hours, the trains were still 120 miles apart. What was the rate of the train?

Answer 1

Okay, so I gave you the systems of equations before, but you need to be able to do it yourself. What are good candidates for variables in this problem?

Answer 2

x

Answer 3

Okay, but what I mean to ask is, what are some unknown values in this word problem. One would be the speed of the slow train... what are some others?

Answer 4

distance

Answer 5

Actually, we do know the distance between them.

Answer 6

oh okay, then the rate

Answer 7

But we don't know how much distance each one has traveled, so that could be one.

Answer 8

Sure, so let's have some variables, \(r_1\) for the rate of the fast train and \(r_2\) for the rate of the slow train.

Answer 9

Now how can we express "The rate of the fast train exceeded the rate of the slow train 10 mph." with the variables \(r_1\) and \(r_2\)?

Answer 10

\[10 + r2 \]

Answer 11

We want an equation with both \(r_1\) and \(r_2\)

Answer 12

But you're very close!

Answer 13

\[r1 = 10 + r2\]

Answer 14

Good. Now, let's have \(t\) be the time that has passed. What does \(t\) equal based on:
"At the end of two hours"

Answer 15

\[t= r1+r2\]

Answer 16

Nope, actually it's just \(t=2\). Alright?

Answer 17

Okay

Answer 18

Finally, we have \(d_1\) as the distance the fast train traveled, and \(d_2\) as the distance the slow train traveled. What algebraic equation can we get from: "Two trains started at the same time from stations which were 360 miles " and " At the end of two hours, the trains were still 120 miles apart."

Answer 19

It won't have \(t\) in it, just \(d_1\) and \(d_2\)

Answer 20

Hint: remember that \(d_1 + d_2\) is going to be how much closer they got to each other.

Answer 21

I don't know, I'm a little confused

Answer 22

is it \[d1 + d2 = 120 ?\]

Answer 23

Okay, the equation is \(d_1+d_2 = 360 - 120 \)

Answer 24

Okay, well 360 - 120 is 240

Answer 25

So the formula we have so far are \[t=2\]\[d_1+d_2 = 240\]\[r_1=r_2+10\]

Answer 26

5 variables, 3 equations, so we need two more equations... we get them from \(d=rt\). So we have \[d_1=r_1t\]\[d_2=r_2t\]

Answer 27

Now we have 5 variables, 5 equations. We can solve it!

Answer 28

Ok, so how do we find d1 and r1?

Answer 29

okay so we use \(t=2\) to plug \(2\) into every other equation that has \(t\) in it.

Answer 30

Okay, d1 = r1(2)

Answer 31

\[d2=r2(2)\]

Answer 32

so now we can substitute \(2r_2\) into the equation with \(d_2\) in it.

Answer 33

We have:\[2r_1+2r_2 = 240\]\[r_1=r_2+10\]Certainly you can solve for them now.

Answer 34

Ok, once again thanks so much, I appreciate it