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Linear Equations, Inequalities and Systems

Lesson 11


Solve linear systems of equations of two variables by substitution.

Common Core Standards

Core Standards


  • 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.

  • N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.

  • A.REI.C.5 — Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

  • A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Foundational Standards


  • 8.EE.C.8

Criteria for Success


  1. Identify the variables, relationship between the variables, and number of functions in system of equations problems. 
  2. Describe why solving by substitution works algebraically. 
  3. Describe the meaning of a solution to a system of linear equations in the context of a problem. 
  4. Describe any domain restrictions that are presented when a system is presented in a contextual situation. 
  5. Describe the equivalence of a solution found graphically and a solution found algebraically. 

Anchor Problems


Problem 1

Two trains leave different cities heading toward each other at different speeds. When and where do they meet? 

Train A, traveling 70 miles per hour (mph), leaves Westford heading toward Eastford, 260 miles away. At the same time, Train B, traveling 60 mph, leaves Eastford heading toward Westford. When do the two trains meet? How far from each city do they meet? 

Guiding Questions

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Mr. Orr Is a Two Trains...

Two Trains... by Jon Orr is made available on Mr. Orr is a under the CC BY 4.0 license. Accessed Oct. 19, 2017, 4:08 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

$${6x+3y=27}$$, find $$x$$ when $$y=x+4$$

What is the value of $$y$$

Guiding Questions

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Problem Set


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

Target Task


A child has a handful of coins of only dimes and nickels. The total amount of money is $1.10, and there is a total of 14 coins.

Write and solve a system of equations (using substitution). Identify the number of nickels and dimes that the child is holding.