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 t d = (air speed  wind speed) times t  d = (ground speed) times t d = (air speed + wind speed) times t 


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 eliminationbyaddition to solve the equations, we will multiply the top row by 9 and the bottom row by 16 to obtain:
Now, add the two equations:
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.