Conquering 3x3 Systems: Mastering Elimination

Algebra 2 Grades High School 15:54 Video
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Lesson Description

Learn to solve systems of three equations with three variables using the elimination method. This lesson plan provides a step-by-step guide with examples and practice problems.

Video Resource

Solving Systems of 3 Equations 3 Variables

Mario's Math Tutoring

Duration: 15:54
Watch on YouTube

Key Concepts

  • Systems of equations
  • Elimination method
  • Variable elimination
  • Back substitution

Learning Objectives

  • Students will be able to identify and set up systems of three equations with three variables.
  • Students will be able to apply the elimination method to reduce a 3x3 system to a 2x2 system, and then to a single variable equation.
  • Students will be able to use back substitution to find the values of all three variables.
  • Students will be able to verify their solutions by substituting the values back into the original equations.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of systems of equations with two variables. Briefly discuss the geometric interpretation (intersection of lines). Introduce the idea of extending this to three variables (intersection of planes).
  • Video Presentation (15 mins)
    Watch the video 'Solving Systems of 3 Equations 3 Variables' from Mario's Math Tutoring. Encourage students to take notes on the steps involved in the elimination method. Pause at key points to clarify concepts and answer questions.
  • Step-by-Step Breakdown (15 mins)
    Outline the general steps for solving 3x3 systems using elimination: 1. Choose a variable to eliminate. 2. Select two equations and eliminate the chosen variable. 3. Select a *different* pair of equations (one of which must be the equation not used in step 2) and eliminate the *same* variable. 4. Solve the resulting 2x2 system. 5. Use back substitution to find the values of all three variables. 6. Check the solution in all three original equations.
  • Guided Practice (20 mins)
    Work through the first example from the video again, but this time involve the students by asking them what the next step should be. Have them perform the calculations. This provides immediate feedback and reinforces the process.
  • Independent Practice (20 mins)
    Have students work in pairs or individually on the second and third examples from the video. Circulate to provide assistance as needed. Encourage them to check their answers.
  • Wrap-up and Discussion (5 mins)
    Summarize the key steps in solving 3x3 systems. Discuss common mistakes and strategies for avoiding them.

Interactive Exercises

  • Online System Solver
    Use an online system of equations solver to check answers and visualize the solution. Websites like Symbolab or Wolfram Alpha can be used.
  • Error Analysis
    Present students with a problem where an error was made in the elimination process. Have them identify the error and correct it.

Discussion Questions

  • Why is it important to eliminate the same variable when reducing the system?
  • What are some strategies for choosing which variable to eliminate?
  • How can you check your answer to ensure it is correct?
  • What are the geometric interpretations of solutions when solving systems of equations?
  • Is there ever a time when a system of three equations with three variables doesn't have one unique solution?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Critical thinking
  • Attention to detail

Multiple Choice Questions

Question 1:

When using the elimination method on a 3x3 system, what is the first step?

Correct Answer: Choose a variable to eliminate.

Question 2:

After eliminating a variable from two pairs of equations, what type of system are you left with?

Correct Answer: A 2x2 system.

Question 3:

What should you do after solving for two of the variables in a 3x3 system?

Correct Answer: Back substitute to find the third variable.

Question 4:

Why is it important to check your solution in all three original equations?

Correct Answer: To make sure you didn't make any mistakes.

Question 5:

What happens geometrically when solving a 3x3 system of linear equations?

Correct Answer: Finding the intersection of planes

Question 6:

Which of the following is NOT a valid operation when using the elimination method?

Correct Answer: Dividing an equation by a variable.

Question 7:

When solving a 3x3 system using elimination, after you combine two equations to eliminate a variable, what must you do next?

Correct Answer: Use at least one of those equations to eliminate the same variable with the unused equation

Question 8:

What does it mean if you arrive at the statement '0 = 5' while solving a system of equations?

Correct Answer: There is no solution to the system.

Question 9:

Which of the following is true when choosing an equation to perform back-substitution with?

Correct Answer: Any equation in the system will result in the correct answer.

Question 10:

For which of the following reasons should you use a check step after finding your final answer?

Correct Answer: Both A and C.

Fill in the Blank Questions

Question 1:

The method used in the video to solve systems of equations is called the ________ method.

Correct Answer: elimination

Question 2:

The goal of the elimination method is to ________ one variable at a time.

Correct Answer: eliminate

Question 3:

After reducing a 3x3 system to a 2x2 system, you solve the 2x2 system using elimination or ________.

Correct Answer: substitution

Question 4:

The process of substituting values back into previous equations to find the remaining variables is called ________ ________.

Correct Answer: back substitution

Question 5:

A solution to a 3x3 system is a set of three numbers that makes all three equations ________.

Correct Answer: true

Question 6:

When solving a system of linear equations with three variables, the solution represents the point where three ________ intersect.

Correct Answer: planes

Question 7:

If you obtain an identity (e.g., 0=0) during the solution process, it indicates that the system has ________ ________ solutions.

Correct Answer: infinitely many

Question 8:

Before starting the elimination process, it's sometimes helpful to ________ the equations to make the coefficients easier to work with.

Correct Answer: rearrange

Question 9:

Always combine at least one new equation each time so you can be sure to use ________ equations.

Correct Answer: all

Question 10:

If eliminating a variable leads to the loss of another variable, it signifies the equation may have been ________.

Correct Answer: dependent

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.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

Teaching Materials

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