Linear Equations, Inequalities and Systems

Lesson 14

Objective

Identify solutions to systems of equations with three variables.

Common Core Standards

Core Standards

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  • 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

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  • 8.EE.C.8

Criteria for Success

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  1. Describe that just like linear systems in two variables, linear systems in three variables do not always have one unique solution. 
  2. Describe the possibilities of a solution in three variables is a line, no solutions, or one solution. 
  3. Show a systematic process (elimination or substitution) for finding the value of each of the three variables. 
  4. Check solution by substituting the solution into all three equations and looking for a true statement. 
  5. Assign variables and write equations that show the relationship between three variables.

Tips for Teachers

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  • Systems of three variables is a hard concept for students to grasp visually. Consider introducing some three-dimensional diagrams to show the possibilities when you have a system of three variables. 
  • For teacher content knowledge, this Khan Academy video from Intro to Linear Systems with 3 Variables is helpful to frame the concept.

Anchor Problems

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Problem 1

What do you notice about the following system of equations? 

$${2x+y-z=2}$$

$${-x-3y+z=-1}$$

$${-4x+3y+z=-4}$$

Guiding Questions

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Problem 2

Below is a word problem and the system of equations that models the context. But, the definition of the variables is missing. Define the variables, solve the system, and give the solution in context of the problem. 

A landlord owns three condominiums: a 1 bedroom condo, a 2-bedroom condo, and a 3-bedroom condo. The total rent she receives is $1,240. She needs to make repairs on the condos, and its costs 10% of the 1-bedroom condo’s rent for its repairs, 20% of the 2-bedroom condo’s rent for its repairs, and 30% of the 3-bedroom condo’s rent for its repairs. The total repair bill was $276. The 3-bedroom condo’s rent is twice the 1-bedroom condo’s rent. How much is the rent for each condo? 

System: 

$${x+y+z=1240}$$

$${.1x+.2y+.3z=276}$$

$${z=2x}$$

Guiding Questions

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References

Diane Ford Word Problems in 3 Variables

Word Problems in 3 Variables by Diane Ford is made available on Summer 2014 Math 80 Intermediate Algebra Course. Accessed Oct. 31, 2017, 1:11 p.m..

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.

Target Task

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Solve the system of equations: 

$${x+3y+z=10}$$

$${x+y+z=2}$$

$${y-2z=2}$$

References