Solving a distance, rate, time problem using a system of linear equations

Question

Two buses leave towns 354 kilometers apart at the same time and travel toward each other. One bus travels 17 kmh slower than the other. If they meet in 2 hours, what is the rate of each bus?

Answer 1

d = st (where s = speed, d = distance, t = time)

for bus 1 (from A at 0km to B at 354km) for 2 hrs

(1) d₁ = s₁(2) = 2s₁

for bus 2 (from B at 354km to A at 0km) for 2 hrs

(2) d₂ = s₂(2) = 2s₂

and we know (because they start at same time, and head towards each other)

(3) d₂ = 354 - d₁

and we know (because we are told one bus is 17kph slower than the other)

(4) s₂ = s₁ + 17

hence we can substitute (3) and (4) in (2) to eliminate d₂ and s₂

(5) (354 - d₁) = 2(s₁ + 17) = 2s₁ + 34

now we can use (1) to substitute and eliminate d₁ to get

(354 - 2s₁) = 2s₁ + 34

=> 354 - 34 = 4s₁

=> s₁ = 320 / 4 = 80

from (4) s₂ = s₁ + 17 = 80 + 17 = 97

hence we have one bus with speed of 80 kph and the other bus with speed of 97 kph

Double check: if bus 1 is 80 kph it travels 160 km in 2 hours, while if bus 2 is 97 kph it travels 194 km in same 2 hours. 194 + 160 = 354 km as required.