Word Problems: Distance II (Systems of Equations)
Solving a system of linear equations means that you will be solving two or more equations with two or more unknowns simultaneously. In order to solve distance, rate, and time problems using systems of linear equations, it is necessary to

It is important to understand the terminology used in the problem. First, a head wind implies that the plane is flying against the wind, which causes the plane fly more slowly. A tail wind, on the other hand, means that the plane is flying with the wind and can go at a faster rate of speed. Air speed is the speed of the plane without consideration of the effect of the wind. Ground speed is the resultant, or the sum, of the wind speed and air speed. A cross wind means that the wind is blowing at an arbitrary angle with respect to the plane's direction and is beyond the scope of this lesson.

 head wind tail wind or equivalently
We need to set up a system of two linear equations. Remember that
distance (d) = rate (r) times time (t).

We need to adjust this formula for consideration of head winds and tail winds as follows:
 d = (ground speed) times td = (air speed - wind speed) times t d = (ground speed) times td = (air speed + wind speed) times t
We will now substitute a variable for air speed (x) and a variable for wind speed (y):
 d = (x - y) times t d = (x + y) times t

Suppose it takes a small airplane flying with a head wind 16 hours to travel 1800 miles. However, when flying with a tail wind, the airplane can travel the same distance in only 9 hours. Find the rate of speed of the wind and the air speed of the airplane.

The first sentence of the problem states: It takes a small airplane flying with a head wind 16 hours to travel 1800 miles. Therefore, we have the following equation:

The second sentence of the problems states: However, when flying with a tail wind, the airplane can travel the same distance in only 9 hours. Therefore, our second equation is the following:

We are ready to solve the following system of equations:

First we will distribute 16 and 9 to obtain:

Using the method of elimination-by-addition to solve the equations, we will multiply the top row by 9 and the bottom row by 16 to obtain:

Now we solve for x:

We have determined that the air speed for the small airplane is 156.25 miles per hour. Substituting into the second equation of the original system to find y, we obtain the following:

Simplifying, we have:

We have now determined that the speed of the wind is 43.75 miles per hour.

Checking our solutions in each equation we have the following:

The solution checks in both equations, therefore, we have determined that the average rate of speed of the airplane for the 1,800 mile trip is 156.25 miles per hour and the rate of speed of the wind is 43.75 miles per hour.

Let's Practice

 Question #1 An airplane flying with a head wind traveled 1000 miles from one city to another in 2 hours and 12 minutes. On the return flight, flying with a tail wind, the total time was only 2 hours. Find the air speed of the plane and the speed of the wind.

 Question #2 A swimmer can swim 18 miles downstream in a nearby river in 3 hours. However, the return trip upstream takes him 6 hours. Find the swimmer’s average speed in still water and find the speed of the river’s current.

Try These
 Question #1 A boat travels upstream for 32 miles in 2 hours. The return trip at the same constant speed with the same current only takes 1 hour and 36 minutes. What is the speed of the boat and the current? A. boat's speed = 18 mph; current = 2 mph B. boat's speed = 2 mph; current = 18 mph C. boat's speed = 19.76 mph; current = 3.77 mph D. boat's speed = 18 mph; current = 34 mph

 Question #2 Two airplanes fly in different directions from the same airport. The second plane leaves a half-hour after the first. The second plane travels at a rate of 60 miles per hour faster than the first. Find the air speed of each airplane if two hours after the first plane starts, the two planes are 2015 miles apart. A. speed of plane #1 = 610 mph; speed of plane #2 = 550 mph B. speed of plane #1 = 601.43 mph; speed of plane #2 = 541.43 mph C. speed of plane #1 = 550 mph; speed of plane #2 = 610 mph D. speed of plane #1 = 575.71 mph; speed of plane #2 = 635.71 mph

As you can see, this type of problem requires carefully setting up two equations with two unknown values. You must be familiar with the formula for distance, rate, and time. You must also be familiar with the distance formulas to use when considering the effect of the speed of the current on the boat. Following the writing of the two equations, you must carefully eliminate one of the variables and solve for the other. Then upon completion of the problem, you must substitute carefully into the two equations to check your answers.

D Saye

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