Systems of Linear Equations

Lesson 3

Math

Unit 6

8th Grade

Lesson 3 of 11

Objective


Classify systems of linear equations as having a unique solution, no solutions, or infinite solutions.

Common Core Standards


Core Standards

  • 8.EE.C.8.A — Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
  • 8.EE.C.8.B — Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Criteria for Success


  1. Understand that a system of linear equations can have one unique solution, no solutions, or infinite solutions
  2. Understand that a system has no solution when the lines are parallel to each other with the same slope and different $$y-$$intercepts; these lines will never intersect and thus never share a common coordinate point.
  3. Understand that a system has an infinite number of solutions when the lines overlap each other with the same slope and $$y-$$intercept; these lines overlap exactly and thus share every coordinate point on the line.
  4. Determine, by inspection of equations, if a system of equations has one, no, or infinite solutions (MP.7).

Tips for Teachers


In this lesson, students inspect equations of systems to determine how many solutions they have. They do not solve systems algebraically yet, as this will come later in the unit. 

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Anchor Problems

25-30 minutes


Problem 1

You and a friend are going for a run along a path. You both run at the same exact speed, but your friend starts 100 yards ahead of the starting point which is where you start. 

The graph below is a sketch of your and your friend’s distance traveled from the starting point over time. When will you and your friend be the same distance from the starting point? Explain your answer using the graph. 

Guiding Questions

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Student Response

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Problem 2

Graph the system of equations below.

$${y={1\over3}(x-6)}$$

$${6y=2x-12}$$

Anika was not sure how to interpret the graph of the system. She thought that since there was only one line visible in the graph then there were no solutions to the system of equations. Do you agree with Anika's reasoning? Explain why or why not. 

Guiding Questions

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Student Response

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Problem 3

Consider the equation $$y = \frac{2}{5} x + 1$$. Write a second linear equation to create a system of equations that has:

a.   Exactly one solution

b.   No solutions

c.   Infinite solutions

Guiding Questions

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Student Response

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References

Illustrative Mathematics How Many Solutions?

How Many Solutions?, accessed on March 10, 2017, 12:03 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem Set

15-20 minutes


Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

5-10 minutes


Determine if each system below has one, no, or infinite number of solutions. Explain your answer.

a.   $${y=3x-2}$$

$${y=3x-3}$$

b.   $${x+2y=-4}$$

$${5x+10y=-20}$$

c.   

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Examples where students are given a system of equations and determine, by inspection, if the system has 1, no, or infinite solutions (Keep the equations simple here so that students can determine their answers by inspection without too much manipulation of the equations.)
  • Examples where students are given graphs of systems (including no and infinite solutions) and determine the solution.

Next

Solve real-world and mathematical problems by graphing systems of linear equations.

Lesson 4
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Analyze & Solve Systems of Equations Graphically

Topic B: Analyze & Solve Systems of Equations Algebraically

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