Lesson Plan: Solving Systems of Equations by Elimination

 

Lesson Summary

This lesson introduces students to solving systems of equations using the elimination method. Over the course of 50 minutes, students will learn to eliminate variables by adding or subtracting equations, allowing them to solve systems more efficiently in certain contexts. Multimedia resources from Media4Math.com are integrated throughout the lesson to provide visual and interactive support. The session concludes with a 10-question quiz, complete with an answer key, to assess understanding.

Lesson Objectives

  • Solve systems of equations using the elimination method.
  • Select and justify appropriate methods (substitution vs. elimination) for solving different systems of equations.

Common Core Standards

  • CCSS.MATH.CONTENT.HSA.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Prerequisite Skills

  • Adding and subtracting linear equations.
  • Simplifying algebraic expressions.

Key Vocabulary

Additional Multimedia Resources

 

 

Warm Up Activities (15 minutes)

1. Desmos Activity: Exploring Elimination Graphically

Students access Desmos to create two linear equations that intersect. Input the following system of equations:

2x + 3y = 120

x - 3y = -60

Step-by-step guide:

  1. Ask students how they can eliminate one of the variables. (Simply add the two equations.)
  2. Students should write down what the sum of the two equations is. (3x = 60)
  3. Have them input this equation and have them share their observations.
  4. Explain that the graph of 3x = 60 intersects the intersection of the system.

 

Systems of Equations

 

2. Hands-On Balancing Scales

In this activity, reinforce to students that the elimination of a variable in an equation, when done correctly, doesn't change the equation. Use a physical analogy of scales to illustrate eliminating quantities from both sides of the scale, while maintaining the balance. Provide manipulatives representing variables (e.g., blocks for x and weights for y). Guide students to balance scales by "adding" or "removing" weights, symbolizing the elimination process.

 

Systems of Equations
Systems of Equations
Systems of Equations

 

3. Collections of Graphs: Identifying Solutions and Relationships

Display a series of images showing graphs of different systems of equations (intersecting lines, parallel lines, coinciding lines).

Group Activity: Divide students into small groups. Each group is assigned one type of graph (e.g., consistent system, no solution, infinite solutions). Students analyze the graphs to determine the type of system and share their findings with the class.

 

Intersecting Lines

Systems of Equations

Parallel Lines

Systems of Equations

Infinite SolutionsSystems of Equations

 

 

Multimedia Resources

 

 

Teach (25 minutes)

Summary

Students learn the elimination method step-by-step, including aligning coefficients, adding or subtracting equations, and solving the resulting single-variable equation. Three detailed examples illustrate the process.

Instructional Examples

Example 1

Solve: 2x + y = 5 and x - y = 1

Step-by-step solution:

  1. Add the equations: 2x + y + x - y = 5 + 1.
  2. Result: 3x = 6.
  3. Solve for x: x = 2.
  4. Substitute x = 2 back: 2(2) + y = 5 to find y = 1.
  5. Solution: (2, 1).

Example 2

Solve: 3x + 2y = 8 and 6x - 2y = 10

  1. Add equations: 3x + 6x + 2y - 2y = 8 + 10.
  2. Result: 9x = 18, x = 2.
  3. Substitute x = 2: 3(2) + 2y = 8.
  4. Solve for y: y = 1.

Example 3: Real-World Application

Compare two rental companies:

  • Company A charges \$50 plus \$2 per mile: y = 2x + 50.
  • Company B charges \$30 plus \$3 per mile: y = 3x + 30.

Step-by-step solution:

  1. Subtract equations to eliminate y: (2x + 50) - (3x + 30) = 0.
  2. Result: -x + 20 = 0.
  3. Solve: x = 20.
  4. Substitute x = 20 into either equation: y = 2(20) + 50 = 90.
  5. Solution: (20, 90). This means at 20 miles, both companies charge \$90.

Multimedia Resources

 

 

Review (10 minutes)

Key Vocabulary Review

  • Elimination: Adding or subtracting equations to remove one variable from a system.
  • Coefficient: The number that multiplies a variable (e.g., in 2x, the coefficient is 2).
  • System of Equations: Two or more equations with the same variables.
  • Solution to a System: A point (or points) that satisfies all equations in the system.

Example 1

Solve: x + 2y = 7 and x - 2y = 1

Solution: (4, 1.5).

Example 2

Solve: 2x + y = 10 and 3x - y = 5

Solution: (3, 4).

Multimedia Resources

 

 

Quiz

Directions: Solve each system of equations. Show your work.

  1. x + y = 5
    x - y = 1

     
  2. y = 2x + 3
    3x + y = 12

     
  3. 4x + y = 10
    2x - y = 2

     
  4. 3x - 2y = 7
    -3x + y = -5

     
  5. A movie rental company charges \$5 per rental plus a \$10 membership fee. Another company charges \$3 per rental plus a \$20 membership fee. Write and solve a system to find when the total costs are equal.

     
  6. 2x + y = 8
    -2x + y = 4

     
  7. y = -x + 4
    2x + 2y = 8

     
  8. A bakery sells muffins for \$2 each and cookies for \$1 each. A customer spends \$10 on 7 items. How many muffins and cookies did they buy?

     
  9. y = 3x - 2
    6x - 2y = 4

     
  10. A gym offers two membership plans. Plan A costs \$30 per month with a \$20 sign-up fee. Plan B costs \$50 per month with no sign-up fee. How many months will it take for the total costs to be the same?

Answer Key

  1. (3, 2)
  2. (3, 2)
  3. (9/5, 33/5)
  4. (1, -2)
  5. 5 rentals
  6. (2, 4)
  7. Infinitely many solutions
  8. 3 muffins, 4 cookies
  9. Infinitely many solutions
  10. 5 months