Jenette rides her bike to the bus station where she barely makes it in time to catch the bus to work. She spends half an hour on her bike and two-thirds of an hour on the bus. If the bus travels 39 mph faster than she travels on her bike, and the total distance from home to work is 40 miles, find the rate of the bike and the rate of the bus.

The rate of the bike is approximately 23.14 mph

The rate of the bus is approximately 62.14 mph

Step-by-step explanation:

The given parameters are;

The time Janette spends on her bike to get to bus station = Half an hour = 0.5 hours = 30 minutes

The time Janette spends on the bus on her way to work = Two-thirds of an hour = 2/3 hours

The speed of the bus = 39 mph + The speed with which Janet travels on her bike

The total distance from her home to her work = 40 miles

To answer the question, let the speed with which Janet travels on her bike in miles per hour = x = the rate of the bike

Therefore, we have;

The speed rate of the bus = 39 + x

Distance = Speed × Time

The distance Janet travels to the bus station = The speed with which Janet travels on her bike × The time Janette spends on her bike to get to bus station

The distance Janet travels to the bus station = x × 0.5 = 0.5·x

The distance travelled by the bus from the bus station to her work = The speed of the bus × The time Janette spends on the bus from the bus station to her place of work

∴The distance travelled by the bus from the bus station to her work = (39 + x) × 2/3 hours = 13 + 2/3·x

The total distance from her home to her work = 40 miles = The distance Janet travels to the bus station + The distance travelled by the bus from the bus station to her work

∴ The total distance from her home to her work = 40 miles = 0.5·x + 13 + 2/3·x

Which gives;

40 = 0.5·x + 13 + 2/3·x

40 = 7/6·x + 13

7/6·x = 40 - 13 = 27

7/6·x = 27

x = 27 × 6/7 ≈ 23.14 mph

The rate of the bike = x ≈ 23.14 mph

The rate of the bike ≈ 23.14 mph

The speed rate of the bus = 39 + x ≈ 39 + 23.14  ≈ 62.14 mph

The speed rate of the bus ≈ 62.14 mph.

you can't solve x because it is 2 linear equation and you only have one equation