A train leaves Danville Junction and travels north at a speed of 75 km/h. Two hours later, a second train leaves on a parallel track and travels north at 125 km/h. How far from the station will they meet?
I have no idea what the system of equations would be for this problem, or how to find the system of equations.
Once I have the system of equations, I should be able to solve it myself.
Thanks! :)
I used to encourage my students to make a chart for these kind of problems
the rows of the chart are the different situations and the columns would be titled
D(distance) R(rate) and T(time)
............D ........R ......T
1st train: 75t ------ 75 ---- t
2nd train: 125(t-2)-- 125 --- t-2
didn't they travel the same distance?
so 125(t-2) = 75t
Notice I only used one variable, if your teacher insists that you use two of them,
define first time as x
the second time as y
then the difference in their times is 2
---> x - y = 2
and the second equation would be 75x = 125y
Train 1 travels for time t at 75 km/hr
Train 2 travels for time (t-2) at 125 km/hr
they both go the same distance
distance = rate * time
therefore
75 t = 125 (t-2)
You see, here's what I did.
I changed the "t" in the equation that both of you gave me (75t=125(t-2)) to an x so I could use elimination to solve the problem.
That made that equation be
125x - 250 = 75x
+
75x = 125 y
That's where my problem comes along... it gets all weird and doesn't work. Is there another different equation I could have for the first equation in x and y form? Because I know that the second equation is basically the same as the first just w/ x's and y's, so is there another totally different equation?
Thanks! :)
You can't just toss around x's and y's indiscriminately.
I gave you the corresponding equations if you have to use x and y
your equation of 75x = 125y contradicts your equation 125x - 250 = 75x
Of course things got weird.
you don't have an equation that states that the difference in their times is 2
Read my alternate solution.
Oh, I kind of skimmed what you said and didn't notice the x-y=2... I'm sorry!
Thanks so much though! :)
P.S: Is the correct answer 375 miles away from the station, meaning x=5 and y=3?
125x - 250 = 75x
+
75x = 125 y
------------------------
but what is y in your system?
You have just written the same fact twice.
To make a system of two equations you need to state two facts
first they go the same distance, at 75 for x hours and the other at 125 for y hours
that is your
75 x = 125 y
now the other fact you are given is
x = y + 2
multiply that by 75 to use elimination
75 x = 75 y + 150
so
0 = 125 y - 75 y -150
150 = 50 y
y = 3
then go back for x
x = y+2 = 5
To solve this problem, you can set up a system of equations based on the information given.
Let's assume that the distance from Danville Junction to the point where the trains meet is "d" kilometers.
Since the first train leaves two hours earlier and travels at a speed of 75 km/h, the distance it covers before the second train starts is 75 * 2 = 150 kilometers.
Now, let's set up the equations:
Equation 1: Distance traveled by the first train + Distance traveled by the second train = Total distance
Equation 2: Time taken by the first train = Time taken by the second train + 2 hours
Using the formula Distance = Speed * Time, we can rewrite the equations as:
Equation 1: 150 + 75t + 125t = d (where t is the time taken by the second train after it starts)
Equation 2: t = (d - 150) / 125
Now we have a system of equations that we can solve.
To find the distance from the station where the trains meet, we need to find the value of "d."
Here's how to solve the system of equations:
1. Substitute the value of t from Equation 2 into Equation 1: 150 + 75((d - 150) / 125) + 125((d - 150) / 125) = d
2. Simplify and solve for d: 150 + (75/125)(d - 150) + (125/125)(d - 150) = d
150 + (3/5)d - (3/5)150 + d - 150 = d
150 + (3/5)d - 3d/5 + d - 150 = d
(8/5)d - (3/5)150 = d
(8/5)d - (3/5)(750/5) = d
(8/5)d - (3/5)(750/5) = (5/5)d
(8d - 3*750)/5 = d
(8d - 2250)/5 = d
3. Multiply through by 5 to clear the fraction: 8d - 2250 = 5d
4. Subtract 5d from both sides: 8d - 5d = 2250
3d = 2250
5. Divide both sides by 3: d = 750
So, the two trains will meet 750 kilometers from Danville Junction.
To double-check your answer, you can substitute the value of "d" back into either Equation 1 or 2 to make sure it satisfies both equations.