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PreAlgebra
Mastering Systems of Equations: The Elimination Method

Mastering Systems of Equations: The Elimination Method

PreAlgebra Grades High School 4:45 Video

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Lesson Description

Learn how to solve systems of linear equations using the elimination method, identify systems with no solution or infinite solutions, and apply this skill to real-world problems.

Video Resource

Solving Systems of Equations Elimination Method

Mario's Math Tutoring

Duration: 4:45

Watch on YouTube

Key Concepts

Elimination method for solving systems of equations
Systems with one solution, no solution, and infinite solutions
Consistent independent, inconsistent, and consistent dependent systems

Learning Objectives

Students will be able to solve systems of linear equations using the elimination method.
Students will be able to identify systems of equations that have no solution or infinite solutions.
Students will be able to classify systems as consistent independent, inconsistent, or consistent dependent based on their solutions.

Educator Instructions

Introduction (5 mins)
Briefly review the concept of systems of equations and their graphical representation. Introduce the elimination method as an alternative to substitution and graphing. Mention that the video will show how to solve using elimination and how to understand when a system has no solution or infinite solutions.
Example 1: Solving a System with One Solution (10 mins)
Watch the video from 0:23 to 1:56. Pause at key steps to explain the process of multiplying equations and eliminating variables. Emphasize checking the solution in both original equations.
Example 2: System with No Solution (8 mins)
Watch the video from 1:56 to 2:54. Discuss the meaning of obtaining a false statement (e.g., 0 = 22) and how it indicates parallel lines and no solution. Introduce the term 'inconsistent system'.
Example 3: System with Infinite Solutions (8 mins)
Watch the video from 2:54 to 3:48. Explain how obtaining a true statement (e.g., 0 = 0) indicates the same line and infinite solutions. Introduce the terms 'consistent dependent system'.
Example 4: Multiplying Both Equations (12 mins)
Watch the video from 3:48 to the end. Explain the process of finding the least common multiple and multiplying both equations to eliminate a variable. Reinforce the importance of checking the solution.
Wrap-up and Practice (7 mins)
Summarize the key concepts and provide students with a couple of practice problems to solve independently. Preview the upcoming quiz.

Interactive Exercises

Solve and Classify
Provide students with several systems of equations. Ask them to solve each system using the elimination method and classify it as consistent independent, inconsistent, or consistent dependent.
Error Analysis
Present students with worked-out solutions to systems of equations that contain errors. Ask them to identify and correct the errors.

Discussion Questions

What are the advantages of using the elimination method over the substitution method?
How can you tell, before solving, whether a system of equations will have one solution, no solution, or infinite solutions?
Can you create a real-world scenario that can be modeled by a system of equations?

Skills Developed

Algebraic manipulation
Problem-solving
Critical thinking
Attention to detail

Multiple Choice Questions

Question 1:

What is the primary goal of the elimination method in solving systems of equations?

To graph the equations

To substitute one variable into another equation

To eliminate one variable by adding or subtracting equations

To find the determinant of the coefficient matrix

Correct Answer: To eliminate one variable by adding or subtracting equations

Question 2:

When solving a system of equations using elimination, if you obtain the statement 0 = 5, what does this indicate?

The system has one unique solution.

The system has infinitely many solutions.

The system has no solution.

The system is consistent and independent.

Correct Answer: The system has no solution.

Question 3:

A system of equations that has infinitely many solutions is classified as:

Inconsistent

Consistent independent

Consistent dependent

Parallel

Correct Answer: Consistent dependent

Question 4:

What should you do after finding a solution (x, y) to a system of equations?

Graph the equations

Substitute the values back into both original equations to check the solution

Solve another system of equations

Cry

Correct Answer: Substitute the values back into both original equations to check the solution

Question 5:

In the elimination method, when is it necessary to multiply one or both equations by a constant?

Always

Only when the coefficients of one variable are opposites

Only when the constant terms are equal

When the coefficients of one variable are not the same or opposites

Correct Answer: When the coefficients of one variable are not the same or opposites

Question 6:

Which of the following describes a 'consistent independent' system of equations?

Parallel Lines

Intersecting Lines

Same Line

Skew Lines

Correct Answer: Intersecting Lines

Question 7:

If after eliminating a variable you obtain 0=0, the system has:

One Solution

Two Solutions

No Solution

Infinite Solutions

Correct Answer: Infinite Solutions

Question 8:

What does an inconsistent system of equations represent graphically?

Intersecting Lines

Overlapping Lines

Parallel Lines

Perpendicular Lines

Correct Answer: Parallel Lines

Question 9:

When using elimination, what operation must be performed after multiplying one or both equations by a constant?

Substitution

Adding

Subtracting

Adding or Subtracting

Correct Answer: Adding or Subtracting

Question 10:

What is the first step in setting up elimination?

Solve one equation for one variable.

Rewrite the equations such that like terms are aligned.

Graph the equations.

Set the determinant of the coefficient matrix to zero.

Correct Answer: Rewrite the equations such that like terms are aligned.

Fill in the Blank Questions

Question 1:

The elimination method aims to __________ one of the variables in the system.

Correct Answer: eliminate

Question 2:

A system with no solution is called __________.

Correct Answer: inconsistent

Question 3:

When a system has infinitely many solutions, the equations represent the __________ line.

Correct Answer: same

Question 4:

If multiplying the equations is necessary, the goal is to make the coefficients of one variable __________ or opposites.

Correct Answer: the same

Question 5:

A consistent dependent system will have __________ solutions.

Correct Answer: infinite

Question 6:

Before adding or subtracting equations in elimination, ensure that like terms are __________.

Correct Answer: aligned

Question 7:

When solving a system, if you arrive at the true statement 0 = 0, the system has __________ solutions.

Correct Answer: infinite

Question 8:

The lowest common multiple is used to determine what constants to __________ each equation by.

Correct Answer: multiply

Question 9:

A system that has a unique solution is said to be consistent __________.

Correct Answer: independent

Question 10:

The process of verifying that your solution is correct involves substituting it back into the __________ equations.

Correct Answer: original

Educational Standards

PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.2:Graphically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

Teaching Materials

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