On the first past of a 317mile trip a salesperson averaged 5

On the first past of a 317-mile trip, a salesperson averaged 58 miles per hour. he averaged only 52 miles per hour on the last part of the trip because of an increased volume of traffic. the total time of the trip was 5 hours and 45 minutes. Find the amount of time at each of the two speed.

Solution

let x = amount of time spent doing 58 miles per hour.
let y = amount of time spent doing 52 miles per hour.
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amount of time spent doing 58 miles an hour plus amount of time spent doing 52 miles per hour equals total number of miles.
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58*x + 52*y = 317
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total time spent is 5 hours 45 minutes which is the same as 5.75 hours since 45 minutes is 3/4 times 60 minutes = 3/4th of an hour = .75 of an hour.
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time spent doing 58 miles per hour plus time spent doing 52 miles an hour equals total time spent.
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x + y = 5.75
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you have 2 equations in 2 unknowns.
they are:
58*x + 52*y = 317
x + y = 5.75
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you can make x = 5.75 - y and substitute in other equation, or
you can solve these simultaneously by multiplying equation 2 by 58 and then subtracting one of the equations from the other one.
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solving as simultaneous equations works as follows:
multiply x+y equation by 58
2 equations become
58*x + 52*y = 317
58*x + 58*y = 58*5.75
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subtracting the second equation from the first gets
0*x + 52*y - 58*y = 317 - 58*5.75
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the x has been eliminated and you are left with
52*y - 58*y = 317 - 58*5.75
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multiplying out and combining like terms gets
-6*y = 317 - 333.5
which becomes
-6*y = -16.5
which becomes
-y = -2.75
multiplying both sides by -1 and it becomes
y = 2.75
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since y = 2.75, and x + y = 5.75, then x = 3.0
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you have
x = 3.0
y = 2.75
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substituting in 58*x + 52*y = 317 gets
58*3.0 + 52*2.75 = 317
which becomes
174 + 143 = 317
which becomes
317 = 317
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answer is
x = 3.0
y = 2.75