Lesson Plan: Applications of Systems of Equations

Lesson Summary

This lesson focuses on applying systems of linear equations to solve real-world problems, including cost comparisons, motion problems, and mixture scenarios. Students will learn to create mathematical models of real-world situations and solve them using substitution, elimination, and graphing techniques. Interactive tools like Desmos and multimedia resources from Media4Math.com will be used throughout the lesson to enhance understanding and engagement. The lesson is designed for a 50-minute class and concludes with a 10-question quiz that includes an answer key.

Lesson Objectives

• Apply systems of equations to solve real-world problems.
• Translate word problems into mathematical models.
• Solve systems of equations using appropriate methods.

Common Core Standards

• CCSS.MATH.CONTENT.HSA.CED.A.2: Create equations in two or more variables to represent relationships.
• CCSS.MATH.CONTENT.HSA.REI.C.6: Solve systems of linear equations exactly and approximately.

Prerequisite Skills

• Solving systems of equations using substitution, elimination, and graphing.
• Translating word problems into equations.

Key Vocabulary

• System of Equations: A set of two or more equations with the same variables.
• Substitution Method: A technique for solving a system of equations by solving one equation for one variable and substituting this expression into the other equation.
• Elimination Method: A method for solving systems of equations by adding or subtracting equations to eliminate one of the variables.
• Graphing Method: Solving a system of equations by graphing the equations on the same set of axes and identifying their point of intersection.
• Intersection Point: The point where two or more graphs meet, representing the solution to a system of equations.
• Break-Even Point: The point at which total cost and total revenue are equal, resulting in neither profit nor loss.
• Solution of a System: The set of values for the variables that satisfy all equations in the system simultaneously.

Additional Multimedia Resources

• Definition: Systems Concepts – Graphical Solutions to Linear Systems https://www.media4math.com/library/22050/asset-preview

• Slide Show of terms related to Systems: https://www.media4math.com/library/slideshow/definitions-systems-equations

• Collection of terms on the topic of Linear Systems: https://www.media4math.com/Definitions--LinearSystems

Warm-Up Activities (10 minutes)

Select from one or more of these activities.

1. Desmos Interactive Activity

Objective: Visualize the solutions to systems of equations dynamically.

Instructions:

1. Open the Desmos Graphing Calculator and input the equations y = 0.5x + 10 and y = 0.3x + 15.
2. Show students how to use sliders for the coefficients and constants in the equations. Demonstrate how changes to these values affect the lines and their intersection.
3. Pose questions such as:
   • "What happens if the slopes of the lines are the same?"
   • "How does changing the y-intercept affect the intersection point?"

Student Interaction: Let students create their own systems of equations and predict the intersection point before verifying it on Desmos.

2. Discussion Prompt

Scenario: Two cell phone companies charge different rates for data usage. Company A charges \$20 flat plus \$0.10 per GB, and Company B charges \$10 flat plus \$0.20 per GB.

Question: "At what point would the total cost for both companies be the same? How can you determine which is more cost-effective beyond that point?"

3. Collaborative Task

Small group discussion: "What does it mean for a business to reach a break-even point or profitability? Why is this important?"

Prompt students to connect this to personal finance or decision-making.

4. Real-Life Connection Scenario

Objective: Demonstrate the practical application of systems of equations.

Scenario: "Two friends are saving money to buy concert tickets. Friend A saves \$5 per week and already has \$10. Friend B saves \$7 per week but hasn’t started saving yet. At what point will they have the same amount of money saved?"

Instructions:

1. Write the equations for the savings plans on the board:
   • y = 5x + 10 (Friend A)
   • y = 7x (Friend B)
2. Ask students to predict when the savings will be equal and what factors might affect their predictions.

Class Discussion: After solving, discuss what the solution means in context.

Teach (30 minutes)

In this section, students will analyze linear system in the context of real-world examples. Each example demonstrates how to translate a word problem into equations and solve them using substitution, elimination, or graphing.

Example 1: Break-Even Analysis

The break-even point is a key concept used to understand when a business covers its costs without making a profit or a loss. Imagine running a lemonade stand: you have expenses like buying lemons, sugar, and cups. The break-even point is the number of cups you need to sell to cover these costs exactly. Selling more than this number means profit, while selling less means a loss. It's a helpful way to figure out how much effort or sales are needed to succeed financially.

1. Introduction to the Activity

Begin by explaining to students that they will watch a video segment titled "Algebra Applications: Systems of Equations – Profit and Loss". Emphasize the real-world relevance of the break-even point and profitability in business contexts. Encourage them to take notes on the examples and explanations provided in the video. Here is the link to the video:

https://www.media4math.com/library/39575/asset-preview

2. Watching the Video

• Ask students to focus on:
  • How systems of equations are used to model revenue and expenses.
  • The role of the break-even point in achieving profitability.
  • The example of Amazon's journey to profitability.

3. Post-Video Engagement

After the video, lead a discussion to unpack the key ideas. Use the following non-mathematical questions to explore broader concepts of profitability and break-even points. Provide answers to guide the discussion.

4. Discussion Questions and Answers

1. Q1: Why is reaching the break-even point crucial for businesses, especially new ones?
   A1: The break-even point marks when revenue equals expenses. For new businesses, reaching this point is a critical milestone that signals the start of potential profitability and long-term viability.
2. Q2: What lessons can businesses learn from the example of Amazon’s early years?
   A2: Persistence and strategic investment are key. Amazon spent years operating at a loss to build its infrastructure and customer base before achieving profitability in 2003.
3. Q3: How might a company’s break-even point influence its pricing and production strategies?
   A3: Understanding the break-even point helps businesses determine the minimum sales volume required and can influence pricing to ensure that production costs are covered efficiently.
4. Q4: How do factors like market demand and competition affect a company’s journey to profitability?
   A4: Market demand dictates the potential sales volume, while competition affects pricing and market share. Both factors must be managed to achieve and sustain profitability.
5. Q5: In what ways do profitability and break-even analysis affect investors’ perceptions of a company?
   A5: Investors view profitability as a sign of financial health. Reaching the break-even point, especially for startups, often boosts investor confidence and can lead to increased stock prices, as seen in Amazon’s case.

5. Summary Activity

Conclude by asking students to reflect on the importance of mathematical models in business decision-making. Have them write a paragraph on how systems of equations provide insights for real-world applications, such as predicting profitability or planning investments. Share reflections in small groups.

Example 2: Cost Comparison

Let’s compare two taxi cab companies, A and B, that provide transportation from the airport to a hotel. The companies have different pricing structures, which are represented by the following equations:

Cost Structures

• Company A: Charges a base fee of \$5 for every trip, plus an additional \$0.50 per mile traveled. The cost equation is:

  y = 0.5x + 5

• Company B: Charges a base fee of \$7 for every trip, plus an additional \$0.30 per mile traveled. The cost equation is:

  y = 0.3x + 7

Break-Even Point

The break-even point is where the costs for both companies are equal. To find this, we solve:

0.5x + 5 = 0.3x + 7

Steps:

1. Subtract 0.3x from both sides: 0.2x + 5 = 7
2. Subtract 5 from both sides: 0.2x = 2
3. Divide by 0.2: x = 10

The break-even point occurs when the distance traveled is 10 miles. At this point, the total cost for both companies is the same.

Detailed Solution

At x = 10 miles:

• Company A: y = 0.5(10) + 5 = 10 dollars
• Company B: y = 0.3(10) + 7 = 10 dollars

Comparison

• For trips less than 10 miles, Company B is cheaper due to its lower per-mile charge, despite a higher base fee.
• For trips more than 10 miles, Company A becomes cheaper because of its lower base fee and higher per-mile charge.

This analysis helps customers decide which taxi cab company is the most cost-effective based on the distance they need to travel.

Example 3: Motion Problem

Problem: Two runners start at different times and travel along the same path. Their distances from the starting point at time t (in hours) are given by the following equations:

• Runner A: d = 5t + 2
  Runner A runs at a speed of 5 miles per hour and has a 2-mile head start at the beginning (t = 0).
• Runner B: d = 6t
  Runner B runs at a speed of 6 miles per hour and starts at the same point as the starting line (d = 0 when t = 0).

When and where will the two runners meet?

Step 1: Set the Equations Equal

To find when the runners meet, we need to determine when their distances (d) are equal. Set the two equations equal to each other:

5t + 2 = 6t

Step 2: Solve for t

Rearrange the equation to isolate t. Subtract 5t from both sides:

2 = 6t - 5t

Simplify:

2 = t

So, t = 2. This means the two runners meet after 2 hours.

Step 3: Find the Distance When They Meet

Now that we know t = 2, substitute this value into either equation to calculate the distance (d) where they meet.

Substitute into Runner A’s equation:

d = 5t + 2

d = 5(2) + 2 = 10 + 2 = 12

Substitute into Runner B’s equation:

d = 6t

d = 6(2) = 12

 Both calculations confirm that the distance is d = 12 miles.

Step 4: Explain the Solution

The two runners meet after 2 hours at a distance of 12 miles from the starting point.