Systems of Equations

Curriculum Alignment

Grade Level: 9th Grade
Subject Area: Mathematics
Curriculum Standard: 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."

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Learning Objectives

By the end of this 20-minute lesson, students will:

1. Grasp the concept of a system of linear equations.
2. Understand three possible outcomes when solving systems of equations: one solution, no solution, or infinitely many solutions.
3. Practice solving an example system using substitution.
4. Relate systems of equations to a real-world scenario.

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Materials Needed

• Whiteboard or large display with markers.
• Graphing calculators (optional but recommended).
• A simple worksheet with systems of equations for students to practice individually (teacher-prepared).
• Colored index cards (2 colors).

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Lesson Outline

1. Warm-Up (2 minutes)

Purpose: Activate prior knowledge and set the stage for new learning.

• Question for the Class: "Imagine two lines on a graph. What are all the ways they could intersect? Discuss!"
  (Students will brainstorm aloud briefly: intersect once, never intersect (parallel), or overlap completely (same line)).
• Use the whiteboard to sketch these three conditions: one intersection point, no intersection, infinitely many intersections.
• Transition statement: "Today, we’re going to connect this visual idea to solving systems of linear equations algebraically."

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2. Introduction to Concept (5 minutes)

Purpose: Define a system of equations and explain outcomes.

• Define System of Equations: "Two or more equations working together in a real-world or abstract problem."
• Briefly explain the three possible outcomes when solving:
  • One Solution: The lines cross at one unique point.
  • No Solution: The lines are parallel and never meet.
  • Infinitely Many Solutions: The two equations describe the same line.

Visualization Challenge: Use two colored index cards—one to represent each equation. Move them to showcase:

• Crossing once (one solution).
• Moving parallel (no solution).
• Lining up perfectly atop each other (infinitely many solutions).

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3. Guided Practice (10 minutes)

Purpose: Walk students through solving a system of equations using substitution.

Example Problem (write on the whiteboard):

Solve the system:

1. y = 2x + 3
2. y = -x + 6

Steps (Go through these step-by-step):

1. Substitute the expression for y from Equation 1 into Equation 2.
   • Write: 2x + 3 = -x + 6
2. Solve for x:
   • Add x to both sides: 3x + 3 = 6
   • Subtract 3 from both sides: 3x = 3
   • Divide by 3: x = 1
3. Find y by substituting x = 1 into Equation 1:
   • y = 2(1) + 3 = 5
4. Conclude: The solution is (1, 5).

• Ask students: “What does this point represent?” (Answer: The point where the two lines intersect!)

Check Work: Substitute x = 1 and y = 5 back into both original equations to verify.

Quick Discussion: What type of solution is this? (One solution.)

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4. Individual Application (5 minutes)

Purpose: Let students try solving a system independently to build confidence.

• Hand out a simple worksheet with the following problem:
  Solve the system:
  1. y = 3x - 4
  2. y = -2x + 1

Instructions: Use substitution to find the solution.

• Circulate and assist as students solve.

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5. Real-World Application Discussion (3 minutes)

Purpose: Help students connect the math to real life.

• Pose a real-world scenario: "Jada and Liam are both saving money. Jada deposits $5 every week (y = 5x + 20), while Liam deposits $10 every week (y = 10x). After how many weeks will they have the same amount in savings?"
• Facilitate a brief discussion without solving—challenge students to finish this scenario at home and come prepared to discuss next class.

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Lesson Closure (Final Minute)

1. Recap key points:

   • What is a system of equations?
   • What are the three possible outcomes when solving systems?
   • How do substitution and graphing help solve them?

2. Encourage students to complete one or two problems from their worksheet to practice substitution as homework.

3. Closing Challenge: “Can you think of another real-life situation where systems of equations might show up? Come ready to share tomorrow!”

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Differentiation

• Struggling Students: Pair them with a peer during guided practice and provide step-by-step cues.
• Advanced Students: Challenge them with a third equation (system of three variables) for bonus practice.
• Hands-On Learners: Let them use graphing calculators to visualize the equations intersecting.

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Assessment

• Informal: Observe participation during guided and independent practice.
• Formal: Monitor worksheet completion for accuracy.

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Note to Teacher: This lesson aims to strike a balance between visual, analytical, and practical learning. Engaging the students with colorful visuals (index cards) and a real-world anchor makes systems of equations less abstract and more accessible.