Curriculum  /  Math  /  11th Grade  /  Unit 1: Linear Functions and Applications  /  Lesson 7

Linear Functions and Applications

Lesson 7

Math

Unit 1

11th Grade

Lesson 7 of 13

Objective

Describe the number of solutions of a system of equations. Verify algebraically.

Common Core Standards

Core Standards

  • 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.
  • 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

  • 8.EE.C.8
  • A.REI.D.10

Criteria for Success

  1. Identify systems of linear equations that have one, no, or all solutions. 
  2. Identify missing values in systems of equations that result in one, no, or all solutions.
  3. Validate the solution to a system of equations algebraically. 
  4. Describe when it is possible for a system of equations to have a solution algebraically but have that solution not be possible in context or graphically. Describe these solutions as "extraneous." 

Tips for Teachers

This lesson includes an introduction to the standard A-REI.2 by identifying extraneous solutions in the context of linear systems.

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Anchor Problems

25-30 minutes

Problem 1

What is the solution $${(x, y)}$$ to the system of equations below? 

$${2y - 5 = 6(x + 2)}$$

$${y + 2 = 3x - 4}$$

Guiding Questions

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Problem 2

Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said, “I wonder whether the dollar belongs inside the cash box or not.” 

The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found $200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not? 

Guiding Questions

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References

Illustrative Mathematics Cash Box

Cash Box, accessed on July 28, 2017, 11:15 a.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.

Target Task

5-10 minutes

In the system of linear equations below, $$a$$ is a constant. 

$$\frac{1}{3}x + 2y =8$$

$$ay= 24 -x$$

  1. If the system has infinite solutions, what is the value of $$a$$?
  2. If the system has one solution, what could be the value of $$a$$? 
  3. What other value would need to change so that this system has no solutions? 

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.

Next

Solve a system of three equations in three variables.

Lesson 8
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Linear Functions

Topic B: Systems of Functions and Constraints

Topic C: Piecewise Functions

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