Word Lesson: 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

• know how to solve a linear equation in terms of one variable: one-step lesson, two-step lesson, multi-step lesson
• know how to solve systems using substitution
• know how to solve systems using elimination-by-addition
• understand that vectors can be added together using head-to-tail vector diagrams

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

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

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, 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:

Equation 1:

Equation 2:

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.