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Lesson Plan: Introduction to Systems of Equations

 

Lesson Summary

This lesson introduces students to the concept of systems of equations and their solutions. Students will learn to identify solutions graphically and determine if a given point is a solution to a system. Multimedia resources from Media4Math.com are integrated throughout the lesson to enhance understanding. The lesson concludes with a 10-question quiz that includes an answer key. This lesson is designed for a 50-minute class period.

Lesson Objectives

  • Define what a system of equations is.
  • Identify solutions to systems of equations graphically.
  • Determine if a given point is a solution to a system of equations.

Common Core Standards

  • CCSS.MATH.CONTENT.HSA.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs).
  • CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of an equation is the set of all solutions.

Prerequisite Skills

  • Graphing linear equations.
  • Finding the slope and y-intercept of a line.
  • Substituting values into an equation.

Key Vocabulary

Additional Multimedia Resources

 

 

Warm Up Activities

Choose from one or more of the activities shown below.

1. Desmos Exploration

Students open the Desmos graphing calculator and input the equations y = 2x + 1 and y = -x + 4.

 

Systems of Equations

 

  1. Guide students to enter each equation. Refer to the screen grab.
  2. Instruct them to observe the graph and identify the point of intersection.
  3. Students click on the intersection point to confirm its coordinates.
  4. Ask students to discuss in pairs why the intersection represents the solution to the system.

2. Visual Introduction

Show this set of images illustrating intersecting and parallel lines. Pose the question: "What do you notice about these lines?"

 

Intersecting Lines

Systems of Equations

Parallel Lines

Systems of Equations

 

3. Graph Paper Hands-On (5 minutes)

Students manually graph y = 2x + 1 and y = -x + 4 on graph paper. Identify the intersection point and share observations with a partner. Here is the graph shown using the Desmos graphing calculator.

 

Systems of Equations

 

 

Teach 

Summary

In this section, students will learn to define and graphically solve systems of equations. Examples include simple systems and real-world scenarios.

Instructional Steps (20 minutes)

  • Define Systems of Equations: Explain that a system of equations is a set of two or more equations with the same variables. Use the example of y = x + 1 and y = -x + 3 to illustrate.

 

Systems of Equations

 

 

  • Graphical Solutions: Demonstrate graphing y = 2x + 1 and y = -x + 4 on a coordinate plane. Highlight the intersection point (1, 3) as the solution to the system.

Systems of Equations

 

Examples

Example 1: Graph y = x + 2 and y = 2x - 1.

  1. Plot y = x + 2 starting with the y-intercept (0, 2) and apply the slope 1 (rise over run).
  2. Plot y = 2x - 1 starting with the y-intercept (0, -1) and apply the slope 2.
  3. Identify the intersection point (3, 5) and verify by substituting x = 3 into both equations.

Systems of Equations

Example 2: Word Problem: Compare two phone plans, y = 0.1x + 20 and y = 0.2x + 10.

In this scenario one plan charges 10 cents a minute and \$20 a month, while the other plan charge 20 cents a minute and a monthly fee of \$10.

  1. Explain that y represents cost, and x represents minutes used.
  2. Graph y = 0.1x + 20 starting at the y-intercept (0, 20) with slope 0.1.
  3. Graph y = 0.2x + 10 starting at the y-intercept (0, 10) with slope 0.2.
  4. Locate the intersection point (100, 30) and interpret: at 100 minutes, both plans cost $30.

 

Systems of Equations

 

Example 3: Graph y = -3x + 5 and y = 2x - 1.

  1. Plot y = -3x + 5 starting at the y-intercept (0, 5) and apply slope -3 (down 3, right 1).
  2. Plot y = 2x - 1 starting at the y-intercept (0, -1) and apply slope 2.
  3. Identify the intersection point (1, 2). Verify by substituting x = 1 into both equations.

Discuss Parallel Lines

Show that y = 2x + 1 and y = 2x - 3 are parallel and have no solution. Reinforce that parallel lines never intersect.

 

Systems of Equations

 

Multimedia Resources

 

 

Review (10 minutes)

  • Recap the definition of a system of equations.
  • Review that the solution is the intersection point of the graphs of the equations.
  • Revisit the key vocabulary and ensure students understand each term.

Activities

  • Quick Review: Verify if (2, 4) satisfies y = x + 2 and y = 2x. Discuss why parallel lines indicate no solution.
  • Partner Practice: Students solve y = x + 1 and y = -x + 5 by graphing and verify the solution.

Multimedia Resources

 

 

Quiz

Directions: Solve each problem below. You may use a graphing calculator.

  1. Identify the solution of y = 3x + 2 and y = -x + 4 from the graph.
     

    Systems of Equations

  2. Determine if (2, 5) is a solution to y = 2x + 1 and y = -x + 5.
     
  3. Graph y = x - 1 and y = -2x + 4. Identify the intersection point.
     
  4. Verify if (0, 3) satisfies y = 3x + 3 and y = x + 3.
     
  5. Explain why parallel lines do not have a solution.
     
  6. Solve y = 2x + 1 and y = -x + 4 graphically.
     
  7. Graph y = -x + 3 and y = x + 1. Find the solution.

  8. Identify the solution to y = 2x + 2 and y = -2x + 8 from the graph.
     

    Systems of Equations
     

  9. Determine if (4, 2) satisfies y = 2x - 6 and y = -0.5x + 4.
     
  10. Why do overlapping lines have infinitely many solutions?
     

Answer Key

  1. (0.5, 3.5)
  2. Yes
  3. (1, 0)
  4. Yes
  5. Since both equations have the same slope but different y-intercepts, they are parallel lines and thus have no solution.
  6. (1, 3). Check students' graphs.
  7. (1, 2)
  8. (1.5, 5)
  9. Yes
  10. Coincident lines represent the same equation and overlap entirely.