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## 30-5x=35-8x?

Elizabeth and her brother are racing home. Elizabeth is running 5mph while her brother is running 8mph. Elizabeth is 30 miles away from home while her brother is 35 miles away from home. When will Elizabeth and her brother meet?

30-5x=35-8x

What is the value for x?

Given the equation: 30 - 5x = 35 - 8x

Solve for x by placing the numbers without x on one side of

the equation and those with x on the other side.

In doing so, remember to change the signs.

For instance, moving 35 to the side of 30 gives (30 - 35)

Likewise, transferring -5x to the side of -8x creates (-8x + 5x)

Notice that the signs change as you move figures across the equal sign

New equation : 30-35 = -8x + 5x

-5 = -3x

x = -5/-3

x = 5/3 hours (5/3 * 60 minutes)

= 1 hour 40 minutes.

Maggie, you have done very well already in setting up the equation correctly. Simply eliminate the variable x at one side of the equation and the number at the other end of the equation. Divide for instant success.

30-5x = 35-8x;

30+3x = 35;

3x = 5;

x = 5/3 (1h40m).

I'm sure you've sensed this solution when you set the equation up; just follow your instincts.

The equation given is correct.

30-5x is Elizabeth's distance from home at time x.  She started 30 miles away and gets 5 miles closer with each passing hour.

35-8x is her brother's distance from home.  When those two are equal, they have met.

30-5x = 35-8x

Add 8x to both sides of the equation:

30+3x = 35

Subtract 30 from each side:

3x = 5

x = 5/3 hours
x = 1 hour and 40 minutes.

You could also solve this a little differently.  They start off 5 miles apart (35-30=5), and the closing speed is the difference of their paces: (8-5=3).

How long does it take to go 5 miles at 3 mph?

x = (5 mi) / (3 mi/hr)

x = 5/3 hr

You had to do exactly the same math operations to solve it this way: Find the difference of 35 and 30, the difference of 8 and 5, and divide the results.  You just used a different analysis for how to approach it.

Hint: Keep the units with the numbers throughout the solution.  Treat the units like variables and some of them will cancel out.  In this case, you would have mi/hr and hr as units.  If you set up the equation correctly, the mi will divide out and only hr will remain (in the numerator!).  Often, the units will tell you when you need to multiply or divide.  If you know the answer needs to be in hours, look at the units on your known quantities and use the operation(s) needed to make the resulting units come out as hours.

This is similar to a problem that you often see on tests that goes something like this:

A bathtub has 60 gallons of water in it.  The spigot is on and is adding water at 2 gallons per minute.  The drain is also open and is draining 4 gallons per minute. When will the tub be empty?  The difference in the two rates is -2 gallons per minute (the water in the tub goes down 2 gallons for each minute elapsed.)  It would take 30 minutes to drain the original 60 gallons.

I don't know if you set up the equation or if it was given to you, but it is useful to think about what x represents and what the equation represents.

As the people run, they get closer to home, so the distance  from home decreases. We know that distance = (rate)(time). 5 and 8 miles per hour are the rates or speeds, so x must be the time run.

Each side of the equation represents how far from home each person is. Elizabeth starts 30 miles away and that distance decreases as time increases.  The brother starts further away, but runs faster. When the two meet, they will be the same distance from home, and that will happen at time x.

Now to solve the equation 30 - 5x = 35 - 8x

I always tell students to get the terms with variables on one side, and the other terms on the other side. I will get all the x terms on the left, and the others on the right.

Add 8x to each side --->   30 + 3x = 35

Subtract 30 from each side --->  3x = 5

Divide by 3 on each side --->        x = (5/3) hours

Convert to minutes ---> (5/3) hours * 60 minutes/hour = 100 minutes = 1 hour and 40 minutes.

You might also wonder how far away from home they are at that time. Just substitute (5/3) hours in for x for either side of the equation:   30 - 5(5/3) = 30 - (25/3) = (90./3) - (25/3) = 65/3 = 21.67 miles away

Maggie,

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

OR

90/3 - 25/3 = 105/3 - 40/3

65/3 = 65/3   Both sides equal

If you would like any further explanation, feel free to contact me! Thanks!