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.
This section can be included in a high-level Algebra I curriculum. It is also available in the Algebra II curriculum, where systems are studied in much more detail.
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
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?
English Words | Translation into Math | Notes/Conventions |
‘Antonio went to see a total of 12 movies’ | $n+d = 12$ |
Note: There are many real-number choices for $\,n\,$ and $\,d\,$ that make this equation true. Here are a few: $$ \begin{gather} \cssId{s32}{0 + 12 = 12}\cr \cssId{s33}{1 + 11 = 12}\cr \cssId{s34}{1.3 + 10.7 = 12}\cr \cssId{s35}{(-2) + 14 = 12}\cr \end{gather} $$Of course, we want whole number solutions, and we also need something else to be true. |
‘... and spent \$86.00’ | $8n + 6d = 86$ |
Each night movie costs \$8.00, so $\,n\,$ night movies cost $\,8n\,$ dollars. Each day movie costs \$6.00, so $\,d\,$ day movies cost $\,6d\,$ dollars. Both $\,8n\,$ and $\,6d\,$ have units of dollars. Also, the number $\,86\,$ has units of dollars. It's important that you have the same units on both sides of the equal sign. Here, we have: dollars plus dollars is dollars. Convention: Write $\,8n\,,$ not (say) $\,8.00n\,$ or $\,\$8n\,$ or $\,\$8.00n\,.$ Note: Convince yourself that there are also infinitely many real-number choices for $\,n\,$ and $\,d\,$ that make this equation true. We want a choice for $\,n\,$ and a choice for $\,d\,$ that make both equations true at the same time. |
Step 3: Choose a Simplest Equation, and Solve For One Variable in Terms of the Other
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$\,\ldots\,$ think I'll choose to solve for $\,n\,$. (It doesn't matter!)
Subtracting $\,d\,$ from both sides, we get: $\,n = 12 - d\,$
Step 4: Use Your New Name in the Other Equation
Substituting $\,n = 12 - d\,$ into the equation $\,8n + 6d = 86\,$ gives:
$$ \cssId{s71}{8(\overset{n}{\overbrace{12 - d}}) + 6d = 86} $$Step 5: Solve the Equation for One Unknown
$8(12 - d) + 6d = 86$ | original equation |
$96 - 8d + 6d = 86$ | distributive law |
$96 - 2d = 86$ | combine like terms |
$-2d = -10$ | subtract $\,96\,$ from both sides |
$d = 5$ | divide both sides by$\,-2\,$ |
Step 6: Use the Known Variable to Find 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:
$n+d = 12$ | the simple equation |
$n + 5 = 12$ | substitute in the known information |
$n = 7$ | subtract $\,5\,$ from both sides |
Step 7: Check, and Report Your Answers
Equations | Check | True? |
$n + d = 12$ | $7 + 5 \,\,\overset{\text{?}}{ = }\,\, 12$ | Yes! |
$8n + 6d = 86$ | $8(7) + 6(5) \,\,\overset{\text{?}}{ = }\,\, 86$ | Yes! (Feel free to use your calculator.) |
The original problem asked how many night movies Antonio attended, so here's what you'd report as your answer:
Antonio attended 7 night movies.
The Good News!
Even though this explanation was very long, you'll actually be writing down very little!
Here's the word problem again, and what I ask my students to write down:
Antonio attended 7 night movies.
Concept Practice
These practice problems will be MORE FUN if they use people you know!
So... take a minute and put in some names!
- Think of a name. Type it in the name box below.
- Is the name you're thinking of male or female? Click the appropriate male/female button.
- Click the ‘Add this name!’ button.
- Put in as many or as few as you want. (We'll throw in some of our own, just to spice things up.)
- Refresh this page if you want to throw everything away and start over.