Lesson Plan: Solving Systems of Equations with No Solution or Infinite Solutions

 

Lesson Summary

This lesson introduces students to systems of equations with no solution or infinitely many solutions. Students will learn to classify systems as consistent or inconsistent and recognize the graphical representation of parallel and coincident lines.

The estimated time to complete the lesson is 50 minutes. Multimedia resources from Media4Math.com will be integrated throughout the lesson to enhance understanding. A 10-question quiz with an answer key will assess student learning.

Lesson Objectives

  • Identify systems of equations with no solution or infinitely many solutions.
  • Classify systems as consistent or inconsistent.
  • Interpret the graphical representation of parallel and coincident lines.

Common Core Standards

CCSS.MATH.CONTENT.HSA.REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with graphs).

Prerequisite Skills

  • Identifying slope and y-intercept.
  • Understanding parallel and coincident lines.

Key Vocabulary

Additional Multimedia Resources

 

 

Warm Up Activities

Choose from among these three activities:

Graphical Exploration with Desmos

Students will use Desmos Graphing Calculator to explore slopes of parallel and overlapping lines. 

  1. Ask students to input equations “y = 2x + 3” and “y = 2x - 4” to observe that the lines are parallel. 
  2. Then input “2x + y = 4” and “4x + 2y = 8” to see coincident lines. 
  3. Discuss the visual differences and ask students to explain their observations.

You can also set up an activity that uses sliders, as shown below.

 

Systems of Equations

 

Discussion Prompt

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

Intersecting Lines

Systems of Equations

Parallel Lines

Systems of Equations

Hands-on Plotting

Students plot two sets of equations on graph paper: one pair of parallel lines and one pair of coincident lines. Discuss findings and compare to graphical software results.

Teach

This set of examples reviews parallel and coincident lines.

Example #1: No Solution (Parallel Lines)

Problem: Solve the system graphically:

y = 2x + 3
y = 2x - 4

Explanation: Both lines have the same slope (2), so they are parallel and will never intersect. Therefore, the system has no solution and is inconsistent.

 

Systems of Equations

 

Example #2: Infinite Solutions (Coincident Lines)

Problem: Solve the system graphically and algebraically:

y = -2x + 4
4x + 2y = 8

Explanation: Notice that the second equation is in standard form. Rewrite it in slope-intercept form:

4x + 2y = 8

2y = -4x + 8

y = -2x + 4

Notice that the second equation is identical to the first. Since the equations are identical, the lines overlap completely. The system has infinite solutions and is consistent.

 

Systems of equations

 

Example #3: Real-World Application

Two food trucks, Truck A and Truck B, sell food and beverages as follows:

  • Truck A sells tacos for $5 each and sodas for $2 each. On average, Truck A makes $2500 a day in total sales.
  • Truck B sells tacos for $10 each and sodas for $4 each. On average, Truck B makes $5000 a day in total sales.

System of Equations (Standard Form)

The system of equations for the two trucks is as follows:

  • For Truck A: 5x + 2y = 2500
  • For Truck B: 10x + 4y = 5000

Converting to Slope-Intercept Form

Rewriting both equations in slope-intercept form (y = mx + b):

  • For Truck A: y = -5/2 * x + 1250
  • For Truck B: y = -5/2 * x + 1250

Thus, the slope-intercept forms are identical:

y = -5/2 * x + 1250

Interpretation of Overlapping Graphs

The slope-intercept equations are identical, so the graphs overlap perfectly. This means:

  • The relationship between the number of tacos sold (x) and the number of sodas sold (y) is proportional across both trucks.
  • The difference lies in pricing and revenue, not in the sales dynamics.
  • The system has infinite solutions.

Domain and Range

The domain (x) is restricted to non-negative integers:

  • For Truck A: 0 ≤ x ≤ 500
  • For Truck B: 0 ≤ x ≤ 500

Thus, the domain is:

x ∈ [0, 500] (whole numbers only)

The range (y) is also restricted to non-negative integers. Solving for y:

y = -5/2 * x + 1250

When x = 0, y = 1250.

When x = 500, y = 0.

Thus, the range is:

y ∈ [0, 1250] (whole numbers only)

Multimedia Resources

 

 

Review

This lesson focused on solving systems of equations that have either no solution or infinitely many solutions. Below is a summary of the key vocabulary used in the lesson:

  • System of Equations: A set of two or more equations with the same variables.
  • Solution of a System: The set of variable values that satisfy all equations in the system simultaneously.
  • Consistent System: A system of equations that has at least one solution.
  • Inconsistent System: A system of equations that has no solution.
  • Independent System: A consistent system with exactly one solution.
  • Dependent System: A consistent system with infinitely many solutions.
  • Parallel Lines: Lines in the same plane that never intersect; they have the same slope but different y-intercepts.
  • Coincident Lines: Lines that lie exactly on top of each other; they have the same slope and the same y-intercept.

Example #1: No Solution (Parallel Lines)

Problem: Solve this system of equations

  • Equation 1: y = 2x + 3
  • Equation 2: y = 2x - 4

Solution:

  1. Identify the slopes and y-intercepts:
    • For y = 2x + 3: Slope = 2, Y-intercept = 3.
    • For y = 2x - 4: Slope = 2, Y-intercept = -4.
  2. Analyze the slopes and y-intercepts:
    • Both lines have the same slope but different y-intercepts.
    • These lines are parallel and do not intersect.
  3. Conclusion:
    • The system is inconsistent and has no solution.

 

Systems of Equations

 

Example #2: Infinite Solutions (Coincident Lines)

Problem: Solve this system.

  • Equation 1: y = -2x + 4
  • Equation 2: 4x + 2y = 8

Solution:

  1. Rewrite the second equation in slope-intercept form:
    • Start with 4x + 2y = 8.
    • Subtract 4x: 2y = -4x + 8.
    • Divide by 2: y = -2x + 4.
  2. Compare the equations:
    • Both equations are y = -2x + 4.
    • They represent the same line.
  3. Conclusion:
    • The system is consistent and dependent, with infinitely many solutions.

 

Systems of Equations

 

Multimedia Resources

Quiz

Solve each system.

  1. Solve the system graphically: y = 3x + 2 and y = 3x - 5.

     
  2. Determine if the system has no solution, infinite solutions, or one solution: 3x + y = 7 and 6x + 2y = 14.

     
  3. Identify the solution type for y = -x + 1 and y = -x + 3.

     
  4. Graph the system and classify it: y = 4x + 6 and y = 4x + 6.

     
  5. Simplify and classify: 5x + 2y = 10 and 10x + 4y = 20.

     
  6. Are the lines y = 0.5x + 2 and y = 0.5x - 3 parallel, coincident, or intersecting?

     
  7. Which term describes the system with the equations y = 2x + 5 and y = -3x + 1?

     
  8. Classify the solution type: y = 3x and y = 3x.

     
  9. Explain why parallel lines do not intersect.

     
  10. What is the solution type for a system where both equations are identical?

     

Answer Key

  1. No solution: The lines are parallel because they have the same slope but different y-intercepts.
  2. Infinite solutions: The equations represent the same line, so every point on the line is a solution.
  3. One solution: The lines have different slopes, so they intersect at exactly one point.
  4. Infinite solutions: The equations simplify to the same line, so they represent the same set of points.
  5. Infinite solutions: The equations are identical after simplification, representing the same line.
  6. No solution: The lines are parallel with the same slope but different y-intercepts, so they never intersect.
  7. Infinite solutions: The equations simplify to the same line, meaning they represent the same set of points.
  8. Infinite solutions: The equations are identical, representing the same line and all points on it.
  9. One solution: The lines have different slopes, so they intersect at exactly one point.
  10. Infinite solutions: The equations simplify to the same line, representing the same set of points.