Lesson Plan: Solving Systems Using Matrices

Lesson Summary

This 50-minute lesson introduces students to solving systems of linear equations using matrices. Students will learn to represent systems of equations as matrices, perform row operations, and use Gaussian elimination to find solutions. The lesson incorporates multimedia resources from Media4Math.com (https://www.media4math.com) and an interactive activity from Desmos.com (https://www.desmos.com) to enhance understanding and engagement. The session concludes with a 10-question quiz, accompanied by a detailed answer key, to assess comprehension.

Lesson Objectives

• Represent systems of linear equations as matrices.
• Perform row operations to simplify augmented matrices.
• Solve systems of equations using Gaussian elimination.

Common Core Standards

CCSS.MATH.CONTENT.HSA.REI.C.8
Represent a system of linear equations as a single matrix equation A⋅X = B. Solve the matrix equation using Gaussian elimination.

Prerequisite Skills

• Understanding of systems of equations.
• Performing basic arithmetic operations.

Key Vocabulary

• Matrix: A rectangular array of numbers arranged in rows and columns.
  • Multimedia Resource: (https://www.media4math.com/library/22188/asset-preview)
• Augmented Matrix: A matrix that includes the coefficients and constants of a system of equations.
• Inverse Matrix: A matrix that, when multiplied by the original matrix, results in the identity matrix. This concept applies to square matrices (matrices with the same number of rows and columns).
• Identity Matrix: A square matrix in which all the elements on the main diagonal are 1, and all the other elements are 0.

Additional Multimedia Resources

• Slide Show of terms related to Systems: https://www.media4math.com/library/slideshow/definitions-systems-equations

• Collection of terms on the topic of Linear Systems: https://www.media4math.com/Definitions--LinearSystems

Warm-Up Activities (10 minutes)

1. Desmos Activity:
   • Go to Desmos Calculator (https://www.desmos.com/calculator).
   • Input the equations 2x + y = 5 and x - y = 1 into the graphing calculator.
   • Identify the solution by locating the point of intersection.
   • Transition to representing these equations in matrix form.
2. Slide Show:
   • Show examples of systems of equations represented as augmented matrices.
   • Discuss the format and how coefficients and constants are organized.
3. Hands-on Puzzle:
   • Provide students with cards showing equations and augmented matrices.
   • Ask them to match equations with their corresponding matrices.

Teach (25 minutes)

This section introduces how to use matrices to solves systems of equations. There are two methods:

1. Augmented Matrices. Write the system as an augmented matrix. Then use row operations to simplify the matrix and find the solutions to the system.
2. Inverse Matrices. Write the system as a matrix of the form A•X = B. Find the inverse matrix A^-1 to solve.

Solve this system using augmented matrices:

x + 2y = 8

3x - y = 5

Step 1: Write the system in augmented matrix form: 

Step 2: Perform row operations to simplify R2:

- Eliminate the 4 in R₂:

- Scale R₂:

Step 3: Perform row operations to simplify R₁:

- Eliminate the 3 in R1:

- Scale R1:

The matrix is now in reduced row echelon form. The solution is:

x = 1, y = 2

Solve this system using augmented matrices:

2x + 3y = 7

-4x + 5y = -3

Step 1: Write the system in augmented matrix form: 

Step 2: Perform row operations to simplify R2:

- Eliminate the -4 in R₂:

- Scale R₂:

Step 3: Perform row operations to simplify R₁:

- Eliminate the 3 in R1:

- Scale R1:

The matrix is now in reduced row echelon form. The solution is:

x = 2 y = 1

Solve this system using augmented matrices:

2x + 3y = 5

4x - y = -7

Step 1: Represent the system in matrix form.

 A•X = B

where: 

Step 2: Compute the inverse of A.

Step 3: Solve for x:

The solution is x = 2 and y = 1.

Multimedia Resources

• Slide Show of terms related to Systems: https://www.media4math.com/library/slideshow/definitions-systems-equations

• Collection of terms on the topic of Linear Systems: https://www.media4math.com/Definitions--LinearSystems

• This slide show provides multiple examples for solving systems of equations using matrices: https://www.media4math.com/library/slideshow/math-examples-solving-systems-equations-using-matrices

Review

Additional Multimedia Resources

• Slide Show of terms related to Systems: https://www.media4math.com/library/slideshow/definitions-systems-equations

• Collection of terms on the topic of Linear Systems: https://www.media4math.com/Definitions--LinearSystems

• This slide show provides multiple examples for solving systems of equations using matrices: https://www.media4math.com/library/slideshow/math-examples-solving-systems-equations-using-matrices

Quiz: Solving Systems Using Matrices

Augmented Matrices Questions

1. Solve the system using augmented matrices:
   x + y = 6
   2x - y = 5
   
    
2. Solve the system using augmented matrices:
   3x + 2y = 14
   4x - y = 11
   
    
3. Solve the system using augmented matrices:
   2x + 3y = 8
   5x - y = 13
   
    
4. Solve the system using augmented matrices:
   x + 3y = 9
   2x - y = 4
   
    
5. Solve the system using augmented matrices:
   x - y = 2
   3x + 2y = 15
   
    

Inverse Matrices Questions

6. Solve the system using inverse matrices:
   x + y = 5
   x - y = 3
   
    
7. Solve the system using inverse matrices:
   2x + y = 8
   3x - y = 10
   
    
8. Solve the system using inverse matrices:
   x + 2y = 10
   3x - y = 7
   
    
9. Solve the system using inverse matrices:
   4x + y = 13
   2x - y = 5
   
    
10. Solve the system using inverse matrices:
    3x + y = 11
    x - 2y = 2

Answer Key

1. x = 4, y = 2
2. x = 3, y = 2
3. x = 3, y = 1
4. x = 2, y = 3
5. x = 5, y = 3
6. x = 4, y = 1
7. x = 4, y = 0
8. x = 3, y = 2.5
9. x = 3, y = 1.5
10. x = 5, y = 2