Introduction

Cramer's Rule is a method for solving linear systems that uses determinants. It was proposed by the Swiss mathematician Gabriel Cramer. It is a very effective algorithm for solving small systems, with up to three variables.

Cramer's Rule can be applied in many areas of mathematics, such as physics, engineering, economics, and more. It can be essential to know in order to solve certain real-world problems, and this method can help to find their solution.

Suggested Resources

• Cramer's Rule from Math is Fun, which concisely and clearly explains what the rule is, how it works, and provides examples.
• Cramer's Rule from Khan Academy, where a video explanation, practice exercises, and printable practice worksheets are available.
• Book: "Matrices and Linear Systems" by Jeff Suzuki, which provides a good theoretical background for Cramer's Rule.

Practice Activity: Unlocking the World with Cramer's Rule

Project Objective

In this project, students will use Cramer's Rule to solve real-world problems that can be represented by systems of linear equations. Students will become proficient in applying Cramer's Rule in order to solve linear systems, while learning about its importance and use within the real world.

Detailed Project Description

Students, in groups of 3-5, will find and analyze real-world examples that can be modeled by a system of linear equations. They will then utilize Cramer's Rule to solve the given system(s) and interpret the results in order to make sense of the problem's solution. Here are the steps students will follow to complete this project:

Phase 1: Identifying the Problem

Students will find real-world situations in which a system of linear equations would need to be solved in order to find the solution. These problems could include anything from determining the cost of items at a store to finding the best route for travel, or even maximizing a project's workflow efficiency. They will need to document the problem that they have chosen to solve, explaining how this problem can be expressed with equations, then write out the linear system.

Phase 2: Applying Cramer's Rule

Once the systems of equations have been created, students will solve those systems using Cramer's Rule.

Phase 3: Interpreting the Results

Once they have solved the systems of equations, students will use the answer(s) they have in the context of the real-world scenario they created. For example, in the context of the problem, what do those values represent?

Materials

• Notebooks, pens, and calculators

Project Deliverables

After completing the project, students will write a report using the following sections:

1. Introduction: This section should describe why the problem(s) chosen can be represented as systems of linear equations, with context about the problem(s) itself.
2. Process: Describe the system(s) of linear equations that were created to model the problem(s), and solve these systems using Cramer's Rule. Document each step of solving the equations thoroughly.
3. Conclusion: Interpret the results of solving the linear systems in the context of the original problem(s). Discuss what the students have learned about solving systems of equations through this experience, and emphasize why this is important and how it is used in everyday problems.
4. References: Provide a list of the sources that aided in this project, such as websites, books, or videos.

The completed report should be given to the instructor a month from the start of the project.