Subject:
Mathematics
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
Algebra, College Algebra, Systems of Equations, Tmm0012
Language:
English
Media Formats:

# Systems of Equations and Inequalities

## Module Overview

This material covers Chapter 7: Systems of Equations and Inequalities chapter of the OpenStax College Algebra Text. This module contains an overview of learning objectives mapped to the OTM state standards, worksheets that correspond to chapter sections, interactive Desmos Activities that pair with the chapter, and a list of supplemental videos that correspond to the chapter content.

Photo by shawn henry on Unsplash

# Section 1: Chapter 7 Introduction and Outcomes

Systems of Equations and Inequalities

A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible?

Let’s consider another question. John received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. John invested$4,000 more in municipal funds than in municipal bonds. He earned \$670 in interest the first year. How much did John invest in each type of fund?

Understanding the correct approach to setting up problems such as these one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving two or three equations and two or three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visual gymnastics.

Chapter Sections

• 7.1 Systems of Linear Equations: Two Variables

• 7.2 Systems of Linear Equations: Three Variables

• 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables

2.Equations and Inequalities:Successful College Algebra students are proficient at solving a wide array of equations and inequalities involving linear, quadratic, higher-order polynomial, rational, exponential, logarithmic, radical, and piecewise-defined functions (including absolute value).

2b.Use correct, consistent, and coherent notation throughout the solution process to a given equation or inequality.

2d.Demonstrate an understanding of the correspondence between the solution to an equation, the zero of a function, and the point of intersection of two curves.*

2f.Solve systems of equations using substitution and/or elimination.

3.Equivalencies:Successful College Algebra students are proficient in creating equivalencies in order to simplify expressions, solve equations and inequalities, or take advantage of a common structure or form.

3a.Purposefully create equivalences and indicate where they are valid.

3b.Recognize opportunities to create equivalencies in order to simplify workflow.

4.Modeling with Functions:Successful College Algebra students should have experience in using and creating mathematics which model a wide range of phenomena.

4a.Interpret the function correspondence and behavior of a given model in terms of the context of the model.

5.Appropriate Use of Technology: Successful College Algebra students are proficient at choosing and applying technology to assist in analyzing functions.

5b.Use technology to verify solutions to equations and inequalities obtained algebraically.

5c.Use technology to obtain solutions to equations to equations and inequalities which are difficult to obtain algebraically and know the difference between approximate and exact solutions.

5d.Use technology and algebra in concert to locate and identify exact solutions.

Chapter Section Objectives and related OTM standards:

7.1 Systems of Linear Equations: Two Variables

• Solve systems of equations by graphing. (2d, 2f, 4a, 5c, 5d)

• Solve systems of equations by substitution. (2d, 2f, 3a, 3b)

• Solve systems of equations by addition. (2d, 2f, 3a, 3b)

• Identify inconsistent systems of equations containing two variables. (2d, 2f, 3a, 3b)

• Express the solution of a system of dependent equations containing two variables. (2d, 2f, 3a, 3b, 5b)

7.2 Systems of Linear Equations: Three Variables

• Solve systems of three equations in three variables. (2d, 2f, 3a, 3b)

• Identify inconsistent systems of equations containing three variables. (2d, 2f, 3a, 3b)

• Express the solution of a system of dependent equations containing three variables. (2d, 2f, 3a, 3b)

7.3 Systems of Nonlinear Equations and Inequalities: Two Variables

• Solve a system of nonlinear equations using substitution. (2d, 2f, 3a, 3b)

• Solve a system of nonlinear equations using elimination. (2d, 2f, 3a, 3b)

• Graph a nonlinear inequality. (2d, 2f, 5b, 5c, 5d)

• Graph a system of nonlinear inequalities. (2d, 2f, 5b, 5c, 5d)

# Section 2: Chapter 7 Worksheets

Please note that this unit only covers section 7.1, 7.2 and 7.3, as the other sections do not fall within the purview of the OTM standards for College Algebra.

The files below are worksheets created to support practice and learning for chatper 7 of the OpenStax College Algebra text.

# Section 3: Chapter 7 Desmos Activites

The links below will direct to pre-built Desmos activities. These can be easily copied, remixed, or compiled into activities for teachers to customize to their class and objectives. For more on how educators can integrate the Desmos activities into their class please click here to link to the Desmos teacher support site:

Below is a short explanation of the Desmos activates linked.

# Linear Systems Bundle

The Linear Systems Bundle contains a large collection of activities that are appropriate for this section. The activities that apply directly to this section are listed below:

Polygraph: Linear Systems
This Custom Polygraph is designed to spark vocabulary-rich conversations about systems of linear equations. Key vocabulary that may appear in student questions includes: parallel, intersect, solution, quadrant, axis, vertical, horizontal, slanted, increasing, and decreasing. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language. Then ask them to play several more games, putting that precise language to work.

Systems of Two Linear Equations
This Custom Polygraph is designed to spark vocabulary-rich conversations about systems of linear equations. Key vocabulary that may appear in student questions includes: parallel, intersect, solution, quadrant, axis, vertical, horizontal, slanted, increasing, and decreasing. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language. Then ask them to play several more games, putting that precise language to work.

Systems of Two Linear Equations
In this activity, students write and solve a system of two linear equations to explore the numerical and graphical meaning of "solution." The activity closes by asking students to apply what they've learned to similar situations.

Solutions to Systems of Linear Equations
This activity will help students understand what it means for a point to be a solution to a system of equations – both graphically and algebraically. This seems like an excellent introduction to the topic.

Playing Catch-Up
Students will develop their understanding of systems of equations, particularly as they're represented as tables, equations, and graphs. They'll apply that understanding to the question, "Will one racer catch another?"

Racing Cars
In this activity, students predict where a pair of cars will meet by using tables, graphs, and/or equations. While students can use any of those representations to solve the challenge, the activity was designed with an eye toward solving systems of linear equations via substitution.

Wafers and Creme
In this activity, students predict which pack of cookies contains more calories. Students then learn the number of calories in each pack and use this new information to calculate the number of calories in a new pack of cookies. This activity can be used either to introduce solving systems via elimination, or as an opportunity to practice those skills.

Card Sort: Linear Systems
In this activity, students practice what they've learned about solving systems of linear equations. The activity begins with a review of the graphical meaning of a solution to a system. Later, students consider which algebraic method is most efficient for solving a given system. Finally, students practice solving equations using substitution and elimination.

Polygraph: Non Linear Systems

his Custom Polygraph is designed to spark vocabulary-rich conversations about systems of non linear equations. Key vocabulary that may appear in student questions includes: parallel, intersect, solution, quadrant, axis, vertical, horizontal, slanted, increasing, and decreasing. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language. Then ask them to play several more games, putting that precise language to work.

# Section 4: Chapter 7 Supplemental Videos

Below is a document that links to supplemental videos for this chapter. Note that these videos are not created by the publisher of the text, so some verbiage or problem solving strategies may vary from what is presented in the text. Please note that these links only covers section 7.1, 7.2 and 7.3, as the other sections do not fall within the purview of the OTM standards for College Algebra.