Linear Equations, Inequalities and Systems

Lesson 1

Objective

Identify the solutions and features of a linear equation and when two linear equations have the same solutions.

Common Core Standards

Core Standards

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  • A.REI.D.10 — Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

  • A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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.7

  • 8.F.A.2

  • 8.F.B.4

Criteria for Success

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  1. Identify that the solutions to a line are only those that are on the graph of the line or can be substituted into a linear equation to produce a true statement. 
  2. Show algebraically that a point is either on the graph of a line or not on the graph of a line. 
  3. Manipulate a linear equation to show that it has the same solutions as an original equation. 

Anchor Problems

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Problem 1

Name at least four points that are solutions to the linear equation $${4x-y=10}$$.

Guiding Questions

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Problem 2

The following two equations have identical solutions. 

$${y=\frac{1}{3}x+4}$$

$${-12=x-3y}$$

Explain how you know that: 

  • The two equations have identical solutions. 
  • The following table of values represents a set of solutions to both equations.

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 that provide the equation and a set of values. Ask students to identify the coordinate points that are solutions to the equation.

Target Task

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Several campuses are holding fundraisers and tallied the amount of tickets they sold, the amount of money they raised, and any one-time donations. 

The prediction for how each campus would price the tickets, plus one-time donations was:  $${y=2.5x + 50}$$

Part A
Which campuses, based on the amount of tickets sold and the total amount raised, appear to have followed the model prediction? 

Part B
Central East and Central West forgot to send in their total amounts of money raised. The director used the prediction equation and his best guess of a sale of 25 tickets and 42 tickets, respectively. How much money did the director predict Central East and Central West raised?