A passenger train leaves the train depot 2 hours after a freight train leaves the same depot. The freight train is traveling 25 mph slower than the passenger train. Find the rate of each train, if the passenger train overtakes the freight train in 2 hours.

• At the point the passenger train overtakes the freight train, they will have travelled the same distance.
• The freight train is travelling 25 mph slower than the passenger train.
• If the passenger train leaves 2 hours after the freight train, and overtakes it in 2 hours, the freight train will have travelled 4 hours and the passenger train will have travelled 2 hours.

Train Type	Rate	Time	Distance
Passenger Train	r	2	d
Freight Train	r – 25	4	d

Now write an equation for each train using d = rt.

Passenger Train
d = r(2)
d = 2r

Freight Train
d = (r – 25)(4) Distribute the 4
d = 4r – 100

You know the distance is the same for both trains. Substitution can be used to solve this system of equations by substituting 2r into the second equation for d and solving for r.

d

=

2r

d

=

4r – 100

2r

=

4r – 100

–4r

 

–4r

=

r

=

50

Passenger Train: 50 mph
Freight Train: r – 25 = 50 – 25 = 25 mph

Michael and Danielle have a combined age of 30. Michael is 2 more than 3 times Danielle’s age. How old are Michael and Danielle?

Let m = Michael’s age and d = Danielle’s age

Translate each sentence into an equation.

Michael and Danielle have a combined age of 30.

m + d = 30

Michael is 2 more than 3 times Danielle’s age.

m = 2 + 3d

In the 2^nd equation, the “m” is already isolated. Using the substitution method, substitute “2 + 3d” in for “m” in the 1^st equation.

m

=

2 + 3d

m + d

=

30

(2 + 3d) + d

=

30

4d

=

28

=

d

=

7

Danielle is 7 years old. Now substitute Danielle’s age into one of the equations to find Michael’s age.

m = 2 + 3d
m = 2 + 3(7)
m = 2 + 21
m = 23

Michael is 23 years old.