| 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
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| 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 |
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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, 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.
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Let's Practice
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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.
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D Saye
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