Mastering Row Reduced Echelon Form: Solving Systems of Equations

Algebra 2 Grades High School 8:14 Video
Print Lesson Plan Watch Video

Lesson Description

Learn how to solve systems of linear equations using the Gauss-Jordan Elimination method and Row Reduced Echelon Form (RREF). This lesson covers writing augmented matrices, applying row operations, and interpreting the solution.

Video Resource

Row Reduced Echelon Form

Mario's Math Tutoring

Duration: 8:14
Watch on YouTube

Key Concepts

  • Augmented Matrix
  • Row Operations (interchanging rows, multiplying by a constant, adding rows)
  • Row Reduced Echelon Form (RREF)

Learning Objectives

  • Students will be able to convert a system of linear equations into an augmented matrix.
  • Students will be able to apply row operations to transform a matrix into Row Reduced Echelon Form.
  • Students will be able to identify the solution to a system of equations from its Row Reduced Echelon Form.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of solving systems of linear equations using methods like substitution and elimination. Briefly introduce the limitations of these methods for larger systems and motivate the need for a more systematic approach like using matrices and Row Reduced Echelon Form.
  • Augmented Matrices (10 mins)
    Explain what an augmented matrix is and how to construct one from a system of linear equations. Show examples of converting systems of equations (2x2 and 3x3) into their corresponding augmented matrices. Emphasize the importance of maintaining the correct order of variables.
  • Row Operations (15 mins)
    Introduce the three elementary row operations: (1) interchanging two rows, (2) multiplying a row by a non-zero constant, and (3) adding a multiple of one row to another row. Explain the purpose of these operations: to transform the matrix into Row Reduced Echelon Form. Provide clear examples of each operation and explain the notation used to represent them.
  • Row Reduced Echelon Form (RREF) (15 mins)
    Define Row Reduced Echelon Form. Explain the conditions that a matrix must meet to be in RREF (leading 1s, zeros above and below leading 1s). Demonstrate how to use row operations to transform a matrix into RREF. Work through Example 1 from the video (0:15-8:23) step-by-step, explaining each row operation and the reasoning behind it.
  • Interpreting RREF (5 mins)
    Explain how to read the solution to the system of equations directly from the RREF matrix. Emphasize that each column represents a variable and the last column represents the constant terms. Show how to write the solution as an ordered triple (x, y, z).
  • Example 2 and Calculator Use (10 mins)
    Work through Example 2 from the video (9:14 onward). If time permits, briefly demonstrate how to use a calculator (TI-83/84) to find the RREF of a matrix (as mentioned in the video). This can be used for checking answers or solving more complex systems.

Interactive Exercises

  • Matrix Transformation Practice
    Provide students with a partially transformed matrix and ask them to perform a specific row operation to achieve a desired outcome (e.g., create a zero in a specific location).
  • RREF Identification
    Present students with several matrices and ask them to identify which ones are in Row Reduced Echelon Form.

Discussion Questions

  • Why is it important to keep track of the row operations you perform?
  • How does Row Reduced Echelon Form make it easier to solve systems of equations?
  • Can you think of situations where solving systems of equations might be useful in real life?

Skills Developed

  • Problem-solving
  • Critical Thinking
  • Procedural Fluency
  • Attention to detail

Multiple Choice Questions

Question 1:

What is the first step in solving a system of equations using Row Reduced Echelon Form?

Correct Answer: Writing the Augmented Matrix

Question 2:

Which of the following is NOT a valid row operation?

Correct Answer: Dividing a row by zero

Question 3:

In Row Reduced Echelon Form, the diagonal elements should ideally be:

Correct Answer: Ones

Question 4:

What does the last column of an augmented matrix represent?

Correct Answer: Constants on the right side of the equations

Question 5:

If a row in RREF is all zeros, what does it indicate?

Correct Answer: The system has infinitely many solutions

Question 6:

What is the primary goal of using row operations?

Correct Answer: To simplify the matrix into RREF

Question 7:

What is the order of the following matrix? [[1, 2, 3], [4, 5, 6]]

Correct Answer: 2x3

Question 8:

In an augmented matrix, what separates the coefficient matrix from the constant terms?

Correct Answer: A vertical line

Question 9:

What does RREF stand for?

Correct Answer: Row Reduced Echelon Form

Question 10:

Which of the following is an example of a scalar multiplication?

Correct Answer: Multiplying a matrix by a constant

Fill in the Blank Questions

Question 1:

The matrix formed by adding the constant terms to the coefficient matrix is called the _________ matrix.

Correct Answer: augmented

Question 2:

A matrix is in _________ when it has leading 1s and zeros above and below the leading 1s.

Correct Answer: RREF

Question 3:

Multiplying a row by a constant is an example of a _________ operation.

Correct Answer: row

Question 4:

If a system has infinitely many solutions, its RREF will contain at least one row of _________.

Correct Answer: zeros

Question 5:

The numbers in front of the variables in a system of equations are called _________.

Correct Answer: coefficients

Question 6:

The Gauss-Jordan Elimination method aims to transform a matrix into _________.

Correct Answer: RREF

Question 7:

Interchanging two rows in a matrix is a valid _________ operation.

Correct Answer: row

Question 8:

The solution to a system of three variables (x, y, z) is written as an ordered _________.

Correct Answer: triple

Question 9:

A matrix is a rectangular array of numbers arranged in rows and _________.

Correct Answer: columns

Question 10:

Adding a multiple of one row to another is a valid _________ operation

Correct Answer: row

Educational Standards

A2.A.1.7:Represent and evaluate mathematical models using systems of linear equations with a maximum of three variables. Graphing calculators or other appropriate technology may be used. A2.N.2:Extend the understanding of numbers and operations to matrices. A2.N.2.1:Use matrices to organize and represent data. Identify the order (dimension) of a matrix.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans