Simple Word Problems Resulting In Linear Equations

Many word problems, upon translation, result in two equations involving two variables (two "unknowns"). In mathematics, a collection of more than one equation being studied together is called a system of equations.

The systems in this section are fairly simple, and can be solved by substituting information from one equation into the other. The procedure is illustrated in the following example:

STEP 1: Name Your Unknowns

Decide what piece(s) of information are unknown, and give name(s) to these things.

Choose names that help you to remember what they represent!

Let n be the number of night tickets (evening shows).

Let d be the number of day tickets (matinee shows).

STEP 2: What Can We Write Down that is True?

Re-read the word problem.

The English words will translate into mathematical sentences involving your unknowns.

You may need additional mathematical concepts in making your translation.

STEP 3: Choose a Simplest Equation, and Solve for One Variable in Terms of the Other

Remember that to solve for a variable means to get it all by itself, on one side of the equation, with none of that variable on the other side.

Here, you are getting a new name for one of your variables that is helpful for finding the solution.

Clearly, the equation $n+d=12$ is simpler than $8n+6d=86$.

We could solve the equation $n+d=12$ for either $n$ or $d$;

hmmm…… think I will choose to solve for $n$. (It does not matter!)

Subtracting dd from both sides, we get: $n=12−d$

STEP 4: Use Your New Name in the Other Equation

Take your new name from the previous step, and substitute it into the remaining equation.

This will give you an equation that has only one unknown.

Substituting $n=12−d$ into the equation $8n+6d=86$ gives: $8\overset{n}{\overbrace{(12-d)}}+6d=86$

STEP 5: Solve the Equation for One Unknown

Solve the resulting equation in one variable. Be sure to write a nice, clean list of equivalent equations.

STEP 6: Use the Known Variable to Find the Remaining Variable

Go back to the simplest equation, substitute in your new information, and solve for the remaining variable.

Make sure you understand the logic being used:

If both $n+d=12$ and $8n+6d=86$ are true, then d must equal 5.

Substitute $d=5$ into the simple equation $n+d=12$ and solve:

STEP 7: Check and Report Your Answers

Check that the numbers you have found make both of the equations true.

Then, report your answer(s) using a complete English sentence.

The original problem asked how many night movies Antonio attended, so here is what you would report as your answer:

Antonio attended 7 night movies.

The Good News!

Even though this explanation was very long, you will actually be writing down very little!

Here is the word problem again, and what I ask my students to write down:

Antonio Attended 7 Night Movies

Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/simple_word_probs.htm
This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License.