To start, the question asks when will they meet, indicating we are solving for time. Since hours are the unit of time used (5 and 8 miles per hour), let x be the number of hours for which each individual has run.
Since Elizabeth is running at 5 miles per hour, she is running at 5x. This indicates for every hour she runs (x), she will travel 5 miles. Since her brother is running at 8 miles per hour, he is running at 8x.
Since Elizabeth started 30 miles away from home and she gets 5 miles closer to home every hour, the equation to identify how close she is to home is: 30 - 5x. Her brother started 35 miles away and is running 8 miles per hour, so his distance from home is: 35 - 8x. e.g. If her brother ran 1 hour (x = 1), he is 35 - 8(1) = 27 miles away from home.
To find the point where Elizabeth and her brother meet, their sum of their equations must be equal, or they must be the same distance away from home. To find this point, you must set the equations equal to each other such that:
30 - 5x = 35 - 8x
You'll notice there is a real number on each side of the equation (30 & 35) and there is a variable x on each side of the equation (5x & 8x). To balance them, you must move all real numbers to one side and all x's to the other side. First, subtract 30 from both sides, which leaves: - 5x = 5 - 8x.
Then, add 8x to both sides, which leaves 3x = 5
Finally, divide both sides by 3, leaving x = 5/3, or 1 & 2/3 hours.
Since there are 60 minutes in an hour, 2/3 of an hour is (2/3 x 60) = 40 minutes. The answer could also be expressed as 1 hour and 40 minutes.
To test your answer, plug in 5/3 for x to ensure both sides are equal, which indicates they are they same distance from home.
30 - (5 x 5/3) = 35 - (8 x 5/3)
30 - 25/3 = 35 - 40/3
At this point, you can turn 25/3 and 40/3 into decimals or you can convert 30 and 35 into thirds.
30 - 8.33 = 35 - 13.33
21.67 = 21.67 Both sides equal
90/3 - 25/3 = 105/3 - 40/3
65/3 = 65/3 Both sides equal
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