Systems of Equations

Grade Level

9th Grade

Duration

55 minutes

Class Size

15 students

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Standards Alignment

Common Core State Standards - Mathematics:

• 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.”

• CCSS.MATH.CONTENT.HSA.REI.C.7
  “Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.” (The focus here will be on linear systems, introducing algebraic and graphical solving.)

• CCSS.MATH.CONTENT.HSA.REI.B.3
  “Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.”

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Learning Objectives

By the end of this lesson, students will be able to:

1. Graphically solve systems of two linear equations by interpreting the point of intersection.
2. Use the substitution method to solve systems of two linear equations algebraically.
3. Verify whether a given solution satisfies both equations in the system.
4. Understand when systems have one solution, no solution, or infinitely many solutions.
5. Apply systems of equations to solve real-world problems involving two variables.

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Materials Needed

• Graph paper (one sheet per student)
• Graphing calculators or online graphing tool (optional)
• Whiteboard, markers
• Printed system of equations problems for substitution practice
• Student notebooks and pencils

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Lesson Outline

1. Introduction (7 minutes)

• Engage: Begin with a real-world contextual problem that can be modeled by a system of equations, e.g., “Two friends are buying tickets for a concert: one buys 3 adult tickets and 2 child tickets for $45, while the other buys 1 adult ticket and 4 child tickets for $40. How much does each ticket cost?”

• Ask students to brainstorm how they might find the prices of the tickets, guiding them toward setting up equations.

• Briefly introduce that today’s lesson will focus on two methods to solve systems of linear equations: graphing and substitution.

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2. Direct Instruction: Graphing Method (12 minutes)

• Explain:

  • Define a system of linear equations (two equations with two variables).
  • Demonstrate graphing each equation on the same coordinate plane.
  • Show how the point of intersection represents the solution (x, y) that satisfies both equations.
  • Discuss cases: one solution (intersecting lines), no solution (parallel lines), infinitely many solutions (same line).

• Model: Using the whiteboard, graph a system step-by-step and identify the solution graphically.

• Check for Understanding: Ask concept questions like “What does it mean if the lines don’t intersect?” or “How can you tell from the graph if there’s only one solution?”

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3. Guided Practice: Graphing (10 minutes)

• Hand out graph paper and provide 2-3 simple systems for students to graph independently or in pairs.
• Circulate the room to provide support, ensuring they plot points accurately and identify the solution point correctly.
• Encourage students to explain verbally or write their understanding of the solution.

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4. Direct Instruction: Substitution Method (10 minutes)

• Explain:
  • Introduce substitution as a method where you solve one equation for one variable and substitute that expression into the other equation.
  • Show step-by-step how to substitute and solve for the remaining variable, then back-substitute to find the other variable.
• Model: Solve the same system from the earlier example algebraically by substitution on the whiteboard.

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5. Guided Practice: Substitution (8 minutes)

• Provide students with 2 new systems of equations to solve using substitution.
• Students work individually while the teacher circulates offering assistance.
• Remind students to check their solutions by plugging values back into the original equations.

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6. Real-World Application and Wrap-up Discussion (5 minutes)

• Present a brief real-world scenario where systems of equations are suitable (e.g., mixing solutions in chemistry, budget planning).

• Have a brief class discussion on how these methods help solve everyday problems.

• Highlight the key takeaways:

  • Multiple methods exist; graphing offers a visual approach, substitution offers algebraic precision.
  • Checking solutions is essential.

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7. Formative Assessment (3 minutes)

• Exit Ticket: Students write the solution of the system and identify the method they would choose to solve a new system. This helps the teacher understand individual student comprehension.

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Differentiation Strategies

• Support: Students struggling with algebraic manipulation may pair with more confident peers during substitution activity or use graphing calculators.
• Challenge: Students finishing early can attempt to solve system problems with no or infinite solutions and explain reasoning.

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Homework Suggestion (Optional)

• Assign students to create their own system of equations based on a scenario they invent and solve it using one method they learned. They can also graph their system.

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Reflection for Teacher

• Monitor student facility with accurately graphing lines and substitution steps.
• Note if students understand relationships between the graphical and algebraic solutions.
• Adjust follow-up lessons to deepen skills, including more complex systems (e.g., quadratic-linear).

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This lesson plan combines active learning with real-world application and multiple methods of problem solving, emphasizing Common Core standards for rigorous understanding of systems of equations.