A motorist drove for 2 hours at one speed and then for 3 hours at another speed. He covered a distance of 252 km. If he had traveled 4 hours at the first speed and one hour at the second speed, he would have covered 244 km. Find the speeds.
2 x + 3 y = 252
4 x + 1 y = 244
multiply second equation by 3
2 x + 3 y = 252
12x + 3 y = 732
-----------------subtract
- 10 x = - 480
x = 48
then y = 244-192 = 52
To find the speeds, let's assume the first speed is x km/h and the second speed is y km/h.
Let's calculate the distance traveled in each case:
Case 1: 2 hours at speed x, then 3 hours at speed y, total distance = 252 km
Distance traveled at speed x = 2 hours * x km/h = 2x km
Distance traveled at speed y = 3 hours * y km/h = 3y km
Total distance = 2x km + 3y km = 252 km
Case 2: 4 hours at speed x, then 1 hour at speed y, total distance = 244 km
Distance traveled at speed x = 4 hours * x km/h = 4x km
Distance traveled at speed y = 1 hour * y km/h = y km
Total distance = 4x km + y km = 244 km
Now we have a system of two equations:
2x + 3y = 252 ...(1)
4x + y = 244 ...(2)
To solve this system, we can use the method of substitution or elimination.
Let's use the method of elimination:
Multiply equation (2) by 3 to make the y coefficients the same:
12x + 3y = 732 ...(3)
Subtract equation (1) from equation (3):
12x + 3y - (2x + 3y) = 732 - 252
10x = 480
x = 48
Substitute the value of x into equation (2):
4(48) + y = 244
192 + y = 244
y = 52
Therefore, the first speed is 48 km/h and the second speed is 52 km/h.