Lesson Plan: Solving Systems of Equations by Substitution

 

Lesson Summary

In this 50-minute lesson, students will learn to solve systems of equations algebraically using the substitution method. They will explore real-world applications to contextualize their learning and utilize multimedia resources from Media4Math.com for guided examples. The lesson includes a warm-up activity, explicit instruction, practice problems, and a 10-question quiz with an answer key to assess understanding.

Lesson Objectives

  • Solve systems of equations using substitution.
  • Interpret the solution of a system of equations within the context of word problems.
  • Develop algebraic reasoning skills by substituting one variable into another equation.

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

  • Solving for a variable in an equation.
  • Substituting values into equations.

Key Vocabulary

Multimedia Resources

 

 

Warm Up Activities (10 minutes)

Choose from one or more of these activities to start your lesson.

1.  Interactive Graphing

Objective: Help students visualize the relationship between two linear equations and identify how solving for one variable affects the system.

Steps:

  1. Open the Desmos Graphing Calculator and input and display two linear equations: y = 2x + 3 and y = -x + 5.
  2. Show students how to toggle equations on and off to observe each graph independently.
  3. Use the sliders feature in Desmos to manipulate one equation, such as changing the slope or y-intercept, and ask students to predict how the solution (intersection point) will change.
  4. Challenge students to isolate y in 3x + 2y = 6, input it into Desmos, and compare their graph to the provided equations.
  5. Discuss how isolating a variable in one equation is analogous to the substitution method.

 

Systems of Equations

 

2. Think-Pair-Share

Objective: Encourage collaborative thinking and develop hypotheses about solving systems algebraically.

Steps:

  1. Write a simple system of equations on the board: y = x + 2 and 2x + y = 7.
  2. Ask students: "If graphing is not an option, how might we solve this system algebraically?"
  3. Have students discuss in pairs for 2 minutes and share ideas with the class.
  4. Highlight the substitution method as one possible approach.

 

Systems of Equations.

 

3. Hands-On Activity

Objective: Allow students to manually explore substitution with simple numeric puzzles.

Steps:

  1. Distribute small index cards. On each card, write an equation or part of an equation (e.g., "x + y = 7" on one card and "x = 4" on another).
  2. Hand out the cards randomly and ask students to find their matching partner by solving for the shared variable.
  3. For example, the student holding x + y = 7 should partner with the student holding x = 4, as y = 3 completes the system.
  4. Discuss the solution as a group and introduce how substitution simplifies the process algebraically.

 

 

Teach (30 minutes)

Summary

This section introduces the substitution method, emphasizes its practical uses, and provides three worked examples to solidify understanding.

Step-by-Step Instruction

Define Substitution: Introduce substitution by solving for one variable in one equation and replacing it in the other equation. Emphasize the process of keeping equations balanced.

Example 1

Solve y = x + 3 and 2x + y = 7.

  • Substitute y = x + 3 into 2x + y = 7: 2x + (x + 3) = 7.
  • Simplify: 3x + 3 = 7.
  • Solve for x: x = 4/3.
  • Substitute x = 4/3 into y = x + 3: y = 4/3 + 3 = 13/3.
  • Solution: (4/3, 13/3).

 

Systems of Equations

 

Example 2

Solve the real-world problem of two phone plans. Plan A charges 10 cents a minute and $20 per month. Plan B charges 20 cents a minute and $10 per month. Compare the plans.

  • Plan A: y = 0.1x + 20.
  • Plan B: y = 0.2x + 10.
  • Substitute y from Plan A into Plan B: 0.1x + 20 = 0.2x + 10.
  • Solve for x: x = 100.
  • Find y: y = 0.1(100) + 20 = 30.
  • Solution: The break-even point is (100, 30).

 

Systems of Equations

 

Multimedia Resources

 

 

Review (10 minutes)

Key Vocabulary Recap

  • Substitution
  • System of Equations
  • Solution

Guided Practice

Solve y = 2x - 1 and x + y = 5 as a class.

Reflection

Discuss the advantages of substitution and scenarios where it is most effective.

Multimedia Resources

 

 

Quiz

Directions: Solve each system of equations using substitution. Show all work.

  1. y = x + 4, 3x + y = 12

     
  2. y = 2x - 5, x + y = 7

     
  3. 2x + y = 10, y = 3x + 1

     
  4. y = -x + 6, 4x + y = 14

     
  5. Real-World Problem: Plan A costs y = 5x + 20. Plan B costs y = 10x. When are costs equal?

     
  6. y = x + 1, 2x - y = 7

     
  7. y = 4x - 3, x + y = 5

     
  8. 2x + y = 6, y = -x + 4

     
  9. Real-World Problem: A car rental company offers two plans. Plan A costs y = 25 (flat fee). Plan B costs y = 0.3x + 35 (variable rate). For what mileage are costs the same?

     
  10. y = 3x + 2, x - y = -4

Answer Key

  1. x = 2, y = 6
  2. x = 4, y = 3
  3. x = 1, y = 4
  4. x = 2, y = 4
  5. x = 4, y = 40
  6. x = 4, y = 5
  7. x = 2, y = 3
  8. x = 1, y = 3
  9. x = -33 1/3. Interpretation: Since mileage (x) cannot be negative, this means that Plan A (flat fee) is always cheaper than Plan B for any positive mileage.
  10. x = -1, y = -1