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Applications of Linear Equations

Questions that arise in the real world are usually expressed in words, rather than in mathematical symbols. Problems in which a question is asked and pertinent information is supplied in the form of words are called “word problems” or “story problems”.

For working these problems, we recommended the following systematic procedure:

Step 1. Begin by reading the problem carefully, several times if necessary, until you understand it well. Draw a diagram whenever possible. Look for the question or questions you are to answer.

Step 2. List all of the unknown numerical quantities involved in the problem. It may be useful to arrange these quantities in a table or chart along with related know quantities. Select one of the unknown quantities on your list, one that seems to play a prominent role in the problem, and call it x. (Of course, any other letter will do as well.)

Step 3. Using information given or implied in the wording of the problem, write algebraic relationships among the numerical quantities listed in step 2. Relationships that express some of these quantities in term of x are especially useful. Reread the problem, sentence by sentence, to make sure you have rewriting all the given information in algebraic form.

Step 4. Combine the algebraic relationship written in step 3 into a single equation containing only x and known numerical constants.

Step 5. Solve the equation for x. Use this value of x to answer the question or questions in step 1.

Step 6. Check your answer to make certain that agrees with the facts in the problem.

"Pump-U-Up" gym charges a $100 enrollment fee, plus$35 per month to be a member. Write an equation that represents the cost as a function of the number of months enrolled. Make a table of values and draw the graph.

y=total cost
x=number of months

The equation that represents the cost as a function of the months enrolled is:

y=100+35x

The table of values is:

 x 0 1 2 3 8 y 100 135 170 205 380

The graph is: A hot air balloon is 200 feet up and descending at a rate of 10 feet every 3 minutes.
a) Write an equation to represent the height of the balloon as a function of time.
b) Sketch a graph.
c) How high will the balloon be after 30 minutes?
d) How long until the balloon reaches the ground?

Solution
a) The equation is
b) The table of values is:

 x 0 12 30 60 y 200 160 100 0

The graph is: c) 100 ft
d) 60 min = 1 hr