Linear Equations, Inequalities and Systems

Lesson 9

Objective

Solve a system of linear equations graphically.

Common Core Standards

Core Standards

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  • A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

  • A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

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  • 8.EE.C.8

Criteria for Success

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  1. Describe that the solution to a system of equations is the intersection point of the lines in the graph or a point that satisfies all equations. 
  2. Use the relationship between variables to write multiple functions that form a system of equations. 
  3. Describe the solution to a system of equations in the context of the problem.

Tips for Teachers

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This lesson builds on work that students have done in grade 8 with systems of linear equations. Solving by graphing is taught first in this series on systems to provide a basis for systems of linear inequalities in the next lesson. 

Anchor Problems

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Problem 1

Draw a system of linear equations with a solution of $${(-1,3)}$$.

Guiding Questions

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Problem 2

You work for a video streaming company that has two monthly plans to choose from: 

  • Plan 1: A flat rate of $7 per month plus $2.50 per video viewed
  • Plan 2: $4 per video viewed

Write and graph an equation for each plan on a coordinate plane. 

Guiding Questions

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References

Illustrative Mathematics Video Streaming

Video Streaming, accessed on Oct. 19, 2017, 3:27 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 The Match Foundation, Inc.

Problem 3

What is the solution to the systems shown below? 

$${\left\{\begin{matrix}y=3x+2 \\ 2y=6x-8 \end{matrix}\right.}$$

Change the system so that it has ALL the same solutions.

Guiding Questions

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Problem Set

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

  • Include problems where the system of equations results in no solutions and others where the system of equations results in all solutions. Do this in context. 

Target Task

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Construct a system of two linear equations where  $${(-2,3)}$$ is a solution to the first equation but not the second equation and where $${(5,-2)}$$ is a solution to the system. 

References