Skip to ContentGo to accessibility page
OpenStax Logo
Algebra 1

1.4.2 Writing Equations to Represent Constraints

Algebra 11.4.2 Writing Equations to Represent Constraints

1.4.2 Writing Equations to Represent Constraints

1.4.2 • Writing Equations to Represent Constraints

Activity

Jada has time on the weekends to earn some money. A local bookstore is looking for someone to help sort books and will pay $12.20 an hour. To get to and from the bookstore on a work day, however, Jada would have to spend $7.15 on bus fare.

1.

Write an equation that represents Jada’s take-home earnings in dollars, EE, if she works at the bookstore for hh hours in one day.

2.

One day, Jada takes home $90.45 after working hh hours and after paying the bus fare. Write an equation to represent this situation.

3.

Is 4 a solution to the last equation you wrote?

  • If so, be prepared to explain how you know it is a solution.
  • If not, be prepared to explain why it is not a solution. Then, find the solution.
4.

Is 7 a solution to the last equation you wrote?

  • If so, be prepared to explain how you know it is a solution.
  • If not, be prepared to explain why it is not a solution. Then, find the solution.
5.

In this situation, what does the solution to the equation tell us?

Are you ready for more?

Extending Your Thinking

Jada has a second option to earn money. She could help some neighbors with errands and computer work for $11 an hour. After reconsidering her schedule, Jada realizes that she has about 9 hours available to work one day of the weekend.

Which option should she choose, sorting books at the bookstore or helping her neighbors? Be prepared to show your reasoning.

Self Check

Jada found a third job option and was offered a position working at a camp for $10.50 per hour. The total bus fare to and from the camp is $8.40. Write an equation that represents how much she will make in a day, d , after working  h hours.
  1. d = 8.40 h + 10.50
  2. d = 10.50 h 8.40
  3. d = 10.50 h + 8.40
  4. d = 10.50 h − 8.40

Additional Resources

Verify a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution of an equation. It is the answer to the puzzle!

How to determine whether a number is a solution to an equation:

Step 1 - Substitute the number in for the variable in the equation.

Step 2 - Simplify the expressions on both sides of the equation

Step 3 - Determine whether the resulting equation is true (the left side is equal to the right side).

  • If it is true, the number is a solution.

  • If it is not true, the number is not a solution.

For example, the equation 2l+2w=722l+2w=72 represents the relationship between the length, ll, and the width, ww, of a rectangle that has a perimeter of 72 units. If we know that the length is 15 units, what is the value of the width?

Step 1 - Substitute the number in for the variable in the equation.

2(15)+2w=722(15)+2w=72

This is an equation in one variable because ww is the only quantity that we don't know. To solve this equation means to find a value of ww that makes the equation true.

Step 2 - Simplify the expressions on both sides of the equation In this case, 21 is the solution because substituting 21 for ww in the equation results in a true statement.

2(15)+2w=722(15)+2w=72

2(15)+2(21)=722(15)+2(21)=72

30+42=7230+42=72

Step 3 - Determine whether the resulting equation is true (the left side is equal to the right side).

  • If it is true, the number is a solution.

  • If it is not true, the number is not a solution.

72=7272=72

Try it

Verify a Solution of an Equation

The equation 2l+2w=482l+2w=48 represents the perimeter of a rectangle.

1.

If the width of the rectangle is 10, what is the new equation?

2.

What is the length of the rectangle that has a width of 10?

3.

Verify the solution.

PreviousNext
Order a print copy
Citation/Attribution
Reuse and redistribution of this content in digital or print format:
  • This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's prior written permission.
  • This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute OpenStax, and must distribute any derivative works under the same license.
  • Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
  • OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
  • Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.
Attribution information
  • If you are redistributing all or part of this book in a noncommercial print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

© Apr 23, 2026 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.