Please tell me the answer I tried everything and still don't get it

Question

A bicyclist bikes the 78 mi to a city averaging a certain speed. The return trip is made at a speed that is 7 mph slower. Total time for the round trip is 19 hr. Find the​ bicyclist's average speed on each part of the trip.

Answer 1

Distance = (Rate)(Time), so Time = Distance / Rate

Let x = speed to the city. Then, x - 7 = speed from the city

(Time to the city) + (Time from the city) = Total time

So, 78 / x + 78 / (x - 7) = 19

78(x - 7) + 78x = 19x(x - 7)

156x - 546 = 19x² - 133x

19x² - 289x + 546 = 0

x = [289 ± √42025] / 38 = [289 ± 205] / 38 = 13 or 2.21

Note that x can't be 2.21 as that would make x - 7 negative.

So, x = speed to the city = 13 mph and speed from the city = 6 mph.

Answer 2

Let's break it down into parts.

So we know that the bicyclist goes between two points, A and B.

Going to the city, the bicyclist starts at point A, ends at point B, and travels 78 miles.

Going from the city to back home, the bicyclist goes from point B to point A and travels back 78 miles.

We know that it takes a time t_A for the bicyclist to make the trip to the city, and it takes a time t_B for the bicyclist to return back home. The total time to go from the city to back home is the sum of these two times. That's our first equation:

t_{A +} t_B = 19 (Equation 1)

Now, we know that the bicyclist goes to the city at a speed v_A. When the bicyclist goes back home, they go at a speed v_B. We also know that going back home, the speed of the bicyclist is 7 miles per hour less than the speed taken when they went to the city. That will be our second equation:

v_B = v_A - 7 (Equation 2)

Finally, the general way to calculate distance is by finding the product of speed and time. In general, the equation is:

d = v * t (Equation 3)

where d is our distance in miles, v is our speed in mph, and t is our time in hours.

We can use this to our advantage. The distance doesn't change for our trips: we go 78 miles into the city, and 78 miles back home. If we isolate time in the equation above, we can use that information for our other two equations:

t_A = 78 / v_A (Equation 4)

t_B = 78 / v_B

t_B = 78 / (v_A - 7) (Equation 5)

Try taking equations 4 and 5 and substituting them into equation 1. You will need to simplify the fractions and use the quadratic formula to find v_A. Once you have v_A, see which answer from the quadratic formula will give you the right times for t_A and t_B to add up to 19 which will also allow you to determine v_B.

Don't give up, you got this!