Systems of Linear Equations

Lesson 11

Math

Unit 6

8th Grade

Lesson 11 of 11

Objective


Model and solve real-world problems using systems of equations.

Common Core Standards


Core Standards

  • 8.EE.C.8.C — Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Foundational Standards

  • 8.EE.C.7
  • 8.F.B.4

Criteria for Success


  1. Write a system of equations to model a situation and interpret the solution in context (MP.4).
  2. Solve systems of equations using any strategy.
  3. Use a graph of a system of equations to see a visual of how different linear equations in a system relate and compare to one another. 

Tips for Teachers


There is only one Anchor Problem for this lesson, which will likely take up a large portion of class. This problem is a great opportunity for students to collaborate with one another and to create visual presentations of their work and strategy to share with others. 

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Anchor Problems


Ivan’s furnace has quit working during the coldest part of the year, and he is eager to get it fixed. He decides to call some mechanics and furnace specialists to see what it might cost him to have the furnace fixed. Since he is unsure of the parts he needs, he decides to compare the costs based only on service fees and labor costs. Shown below are the price estimates for labor that were given to him by three different companies. Each company has given the same time estimate for fixing the furnace.

  • Company A charges $35 per hour to its customers.
  • Company B charges a $20 service fee for coming out to the house and then $25 per hour for each additional hour.
  • Company C charges a $45 service fee for coming out to the house and then $20 per hour for each additional hour.

For which time intervals should Ivan choose Company A, Company B, or Company C? Support your decision with sound reasoning and representations. Consider including equations, tables, and graphs.

Guiding Questions

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References

Illustrative Mathematics Fixing the Furnace

Fixing the Furnace, accessed on March 21, 2018, 8:12 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.

Problem Set

Fishtank Plus Content

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

Target Task


Problem 1

The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions. 

  No solution Exactly 1 solution Infinite number of solutions
$${9=4x+7}$$      
$${y=3y+5}$$      
$${y={x\over2}}$$      

$${3x+6y-9=0}$$

     

$${x=y}$$

$${1.25x=1.25y}$$

     

$${9.9=6x+8y}$$

$${9x=2.5y-8.8}$$

     

$${11x-2y=1.5}$$

$${11x-2y=2.5}$$

     

References

Achieve the Core Simultaneous Linear Equations Mini-AssessmentQuestion #2

Simultaneous Linear Equations Mini-Assessment by is made available by Achieve the Core under the CC0 1.0 license. Accessed March 21, 2018, 8:17 p.m..

Problem 2

Kim has a small container and a large container as shown. 

It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container. 

What is the size of each of the two containers? 

References

Achieve the Core Simultaneous Linear Equations Mini-AssessmentQuestion #12

Simultaneous Linear Equations Mini-Assessment by is made available by Achieve the Core under the CC0 1.0 license. Accessed March 21, 2018, 8:17 p.m..

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.

  • Include spiraled review from the unit.
  • MARS Summative Assessment Tasks for Middle School Chips and Candy
  • Inside Mathematics Performance Assessment Tasks Grades 3-High School Picking Apples
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Lesson 10

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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