Applications Involving Linear Equations

Questions that arise in the real world are usually expressed in words rather than in mathematical symbols. For example: What will the monthly payment on my mortgage be? How much insulation must I use in my house? What flight should I take to Boston? How safe is this new product? In order to answer such questions, it is necessary to have certain pertinent information. For instance, to determine the monthly payment on a mortgage, you need to know the amount of the mortgage, the interest rate, and the time period involved.

Problems in which a question is asked and pertinent information is supplied in the form of words are called “word problems” or “story problems” by students and teachers alike. We study word problems that can be worked by setting up an equation containing the unknown and solving it by the methods illustrated. To solve 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 known 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 the 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 terms of $$x$$ are especially useful. Reread the problem, sentence by sentence, to make sure you have rewritten 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 it agrees with the facts in the problem.



One number is $$15$$ less than a second number. Three times the first number added to twice the second number is $$80$$. Find the two numbers.



We follow the procedure outlined above.

Step 1. Question: What are the two numbers?

Step 2. Unknown quantities: $$the{\text{ }}1st{\text{ }}number$$ and  $$the{\text{ }}2nd{\text{ }}number$$

Step 3. Information given

(i) $$the{\text{ }}1st{\text{ }}number = the{\text{ }}2nd{\text{ }}number – 15$$; that is,
$$x = the{\text{ }}2nd{\text{ }}number – 15$$

(ii) $$3\left( {the{\text{ }}1st{\text{ }}number} \right) + 2\left( {the{\text{ }}2nd{\text{ }}number} \right) = 80$$; that is,
$$3x + 2\left( {the{\text{ }}2nd{\text{ }}number} \right) = 80$$

Step 4. From relationship (i) in Step 3, we have
$$the{\text{ }}2nd{\text{ }}number = x + 15$$

Therefore, relationship (ii) can be written as
$$3x + 2\left( {x + 15} \right) = 80$$

Step 5. Solving the equation $$3x + 2\left( {x + 15} \right) = 80$$, we obtain
$$3x + 2x + 30 = 80$$
$$5x = 50$$
$$x = 10$$

Therefore $$the{\text{ }}1st{\text{ }}number = x = 10$$ and
$$the{\text{ }}2nd{\text{ }}number = x + 15 = 10 + 15 = 25$$

Step 6. Check the answer: $$10$$ is $$15$$ less than $$25$$ and $$3\left( {10} \right) + 2\left( {25} \right) = 80$$.