# Cluster 3: Analyze and solve linear equations and pairs of simultaneous linear equations. (Major Cluster)Archived Export Print

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.8.EE.3
Title: Analyze and solve linear equations and pairs of simultaneous linear equations. (Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Expressions & Equations

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.8.EE.3.AP.7a
Simplify linear equations and solve for one variable.
MAFS.8.EE.3.AP.8a
Identify the coordinates of the point of intersection for two linear equations plotted on a coordinate plane.
MAFS.8.EE.3.AP.8b
Given two sets of coordinates for two lines, plot the lines on a coordinate plane and define the rise/run (m) for each line to determine if the lines will intersect or not.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Educational Games

Solving Equations: Same Variable, Both Sides, One Solution:

In this challenge game, you will be solving equations with variables on both sides. Each equation has a real solution. Use the "Teach Me" button to review content before the challenge. After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Educational Game

## Educational Software / Tool

Free Graph Paper:

A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Type: Educational Software / Tool

## Formative Assessments

How Many Solutions?:

Students are asked to determine the number of solutions of each of four systems of linear equations without solving the systems of equations.

Type: Formative Assessment

Solving System of Linear Equations by Graphing:

Students are asked to solve a system of linear equations by graphing.

Type: Formative Assessment

Identify the Solution:

Students are asked to identify the solutions of systems of equations from their graphs and justify their answers.

Type: Formative Assessment

Linear Equations - 2:

Students are asked to solve a linear equation in one variable with rational coefficients.

Type: Formative Assessment

Linear Equations - 1:

Students are asked to solve a linear equation in one variable with rational coefficients.

Type: Formative Assessment

Equation Prototypes:

Students are asked to write examples of equations with one solution, no solutions, and infinitely many solutions.

Type: Formative Assessment

Counting Solutions:

Students are asked to identify and explain whether given equations have one solution, no solutions, or infinitely many solutions.

Type: Formative Assessment

Writing System Equations:

Students are given word problems and asked to write a pair of simultaneous linear equations that could be used to solve them.

Type: Formative Assessment

System Solutions:

Students are asked to solve a word problem by solving a system of linear equations.

Type: Formative Assessment

Solving Systems of Linear Equations:

Students are asked to solve three systems of linear equations algebraically.

Type: Formative Assessment

Linear Equations - 3:

Students are asked to solve a linear equation in one variable with rational coefficients and variables on both sides of the equation.

Type: Formative Assessment

Solving Real-Life Problems: Baseball Jerseys:

This lesson unit is intended to help you assess how well students are able to:

• Interpret a situation and represent the variables mathematically.
• Select appropriate mathematical methods to use.
• Explore the effects of systematically varying the constraints.
• Interpret and evaluate the data generated and identify the break-even point, checking it for confirmation.
• Communicate their reasoning clearly.

Type: Formative Assessment

## Lesson Plans

How Low Can You Go?:

Land is becoming scarce. With the diminishing amount of viable land available for humans, animals, and vegetation, we must start exploring other options as populations expand. You will investigate the thousands of miles of land below sea-level and create a structure to withstand applied pressure, imitating the pressure felt in deep water.

Students attempt to avoid an "egg-splosion" by creating a model to withstand the pressure felt by objects as if they are at varying depths below sea-level.

Type: Lesson Plan

How Will the Ninja Capture the Valuable Princess?:

This lesson allowed students to solve two step equations involving a mythological story line in which the princess (variable) is protected by a body guard (number added or subtracted in an equation) and by a protector (number multiplied or divided by the variable). The three characters live in a castle, surrounded by the moat (equal sign) and an innocent bystander that lives outside the castle (number on the opposite side of the variable). However, Ninjas are infiltrating the castle in order to steal the "valuable" princess. Using this story line, students must then decide who the Ninja must eliminate first in order to get to the princess. This lesson can also be used to solve equations with like terms on the same side and equations with the same variable on each side.

Type: Lesson Plan

Battle on the High Seas: Appyling Systems of Linear Equations:

This lesson is designed to introduce solving systems of linear equations in two variables by graphing. Students will find the solutions of systems of linear equations in two variables by graphing "paths" of battleships and paths of launched torpedoes targeting them. The solutions of the systems will represent the intersection of the paths of a targeted ship (modeled by a linear equation) and the path of a torpedo from a battleship (modeled by another linear equation).

Type: Lesson Plan

Building and Solving Equations 1:

This 90-minute lesson helps teachers assess how well students are able to create and solve linear equations with one variable and in more than one way. Students will work individually and in pairs on collaborative activities. They will evaluate and sample student work and create linear equations for each other to solve. In order to complete this lesson, students will need copies of the assessment tasks, paper, mini-whiteboards, pens, and erasers.

Type: Lesson Plan

Classifying Solutions to Systems of Equations:

This lesson unit is intended to help you assess how well students are able to classify solutions to a pair of linear equations by considering their graphical representations. In particular, this unit aims to help you identify and assist students who have difficulties in:

• Using substitution to complete a table of values for a linear equation.
• Identifying a linear equation from a given table of values.
• Graphing and solving linear equations.

Type: Lesson Plan

Solving Linear Equations in One Variable:

This lesson is intended to help you assess how well students are able to:

• Solve linear equations in one variable with rational number coefficients.
• Collect like terms.
• Expand expressions using the distributive property.
• Categorize linear equations in one variable as having one, none, or infinitely many solutions.
It also aims to encourage discussion on some common misconceptions about algebra.

Type: Lesson Plan

Solving Real-Life Problems: Baseball Jerseys:

This lesson unit is intended to help you assess how well students are able to interpret a situation and represent the variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, interpret and evaluate the data generated and identify the break-even point, checking it for confirmation and communicate their reasoning clearly.

Type: Lesson Plan

Repeating Decimals:

This lesson unit is intended to help you assess how well students are able to translate between decimal and fraction notation, particularly when the decimals are repeating, create and solve simple linear equations to find the fractional equivalent of a repeating decimal, and understand the effect of multiplying a decimal by a power of 10.

Type: Lesson Plan

A Scheme for Solving Systems:

Students will graph systems of linear equations in slope-intercept form to find the solution to the system, the point of intersection. Because the lesson builds upon a group activity, the students have an easy flow into the lesson and the progression of the lesson is a smooth transition into solving systems algebraically.

Type: Lesson Plan

Company Charges:

In this lesson the students will learn how to write and solve linear equations that have one solution, infinitely many solutions and no solutions. As the students decipher word problems, they will recognize what elements of the equations effect the number of possible solutions. This lesson is guided by a Powerpoint presentation.

Type: Lesson Plan

Where does my string cross?:

Students will graph two linear functions using pieces of string that intersect and discover what the point of intersection has to do with both functions. It will get tricky when the functions do not intersect, or when they transform into the same equation.

Type: Lesson Plan

Determining the density of regular and irregular objects:

This MEA provides students with opportunities to practice solving one-step equations while learning about density. Students will calculate density of regular and irregular objects.

Type: Lesson Plan

Students will learn to find the solutions to a system of linear equations, by graphing the equations.

Type: Lesson Plan

Exploring Systems of Equations using Graphing Calculators:

This lesson plan introduces the concept of graphing a system of linear equations. Students will use graphing technology to explore the meaning of the solution of a linear system including solutions that correspond to intersecting lines, parallel lines, and coinciding lines.
Students will also do graph linear systems by hand.

Type: Lesson Plan

The Variable Stands Alone:

Students will practice and create problems solving linear equations that involve one solution, no solution, infinitely many solutions. There will be class think aloud portions so students can discuss their thoughts. In addition, students will create their own real-world problems that can be used for the next days extension exercise.

Type: Lesson Plan

Classifying Solutions to Systems of Equations:

This lesson unit is intended to help you assess how well students are able to classify solutions to a pair of linear equations by considering their graphical representations. In particular, this unit aims to help you identify and assist students who have difficulties in using substitution to complete a table of values for a linear equation, identifying a linear equation from a given table of values and graphing and solving linear equations.

Type: Lesson Plan

Exploring Systems with Piggies, Pizzas and Phones:

Students write and solve linear equations from real-life situations.

Type: Lesson Plan

Human systems of linear equations:

Students will work in cooperative groups to demonstrate solving systems of linear equations. They will form lines as a group and see where the point of intersection is.

Type: Lesson Plan

Solving Linear Equations in One Variable:

This lesson unit is intended to help you assess how well students are able to:

• Solve linear equations in one variable with rational number coefficients.
• Collect like terms.
• Expand expressions using the distributive property.
• Categorize linear equations in one variable as having one, none, or infinitely many solutions.
It also aims to encourage discussion on some common misconceptions about algebra.

Type: Lesson Plan

## Original Student Tutorials

Multi-Step Equations: Part 5 How Many Solutions?:

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• [CURRENT TUTORIAL] Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-Step Equations: Part 4 Putting it All Together:

Learn alternative methods of solving multi-step equations in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• [CURRENT TUTORIAL] Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-step Equations: Part 3 Variables on Both Sides:

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• [CURRENT TUTORIAL] Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-Step Equations: Part 2 Distributive Property:

Explore how to solve multi-step equations using the distributive property in this interactive tutorial.

This is part two of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• [CURRENT TUTORIAL] Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-Step Equations: Part 1 Combining Like Terms:

Learn how to solve multi-step equations that contain like terms in this interactive tutorial.

This is part one of five in a series on solving multi-step equations.

• [CURRENT TUTORIAL] Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Interpreting the Graph:

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Quinoa Pasta 1:

This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns. This progression of tasks helps distinguish between 8th grade and high school expectations related to systems of linear equations.

Solving Equations:

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Cell Phone Plans:

This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the cell phone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task.

The Sign of Solutions:

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Two Lines:

In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.

Selling Fuel Oil at a Loss:

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table (Standard for Mathematical Practice, ).

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

Example 3: Solving Systems by Substitution:

This example demonstrates solving a system of equations algebraically and graphically.

Type: Tutorial

Substitution Method Example 2:

This video demonstrates a system of equations with no solution.

Type: Tutorial

The Substitution Method:

This video shows how to solve a system of equations using the substitution method.

Type: Tutorial

Checking Solutions to Systems of Equations Example:

This video demonstrates testing a solution (coordinate pair) for a system of equations

Type: Tutorial

Using a Graph to Analyze Solutions to Linear Systems:

This video demonstrates analyzing solutions to linear systems using a graph.

Type: Tutorial

Example of System with No Solution:

This video shows how to algebraically analyze a system that has no solutions.

Type: Tutorial

Solve a Consecutive Integer Problem Algebraically:

This video will show how to solve a consecutive integer problem.

Type: Tutorial

Distributive Property to Simplify Equations:

Use the Distributive Property while solving equations with variables on both sides.

Type: Tutorial

Introduction to solving an equation with variables on both sides:

Students will learn how to solve an equation with variables on both sides. This tutorial shows a final answer expressed as an improper fraction and mixed number.

Type: Tutorial

Application of the Distributive Property to Solve a Multi-Step Equation:

This video shows how to solve the equation (3/4)x + 2 = (3/8)x - 4 using the Distributive Property.

Type: Tutorial

Solving Equations with the Distributive Property:

This video shows how to solve an equation involving the Distributive Property.

Type: Tutorial

Solving a Multi-Step Equation:

This example involves a variable in the denominator on both sides of the equation.

Type: Tutorial

Solving an equation with variables on both sides:

Students will learn how to solve an equation with variables on both sides. Students will also learn how to distribute and combine like terms.

Type: Tutorial

Linear equation word problem:

Learn how to solve a word problem by writing an equation to model the situation. In this video, we use the linear equation 210(t-5) = 41,790.

Type: Tutorial

Solving Equations: Word Problem:

This tutorial shows a word problem in which students will find the dimensions of a garden given only the perimeter. Students will create an equation to solve.

Type: Tutorial

Solving a more complicated equation:

This example demonstrates how to solve an equation expressed in the form ax + b = c.

Type: Tutorial

Solving a two-step equation with a numerator of x:

This video shows how to solve an equation by isolating the variable in the numerator.

Type: Tutorial

Two-Step Equations:

Students will practice two step equations, some of which require combining like terms and using the distributive property.

Type: Tutorial

Solving two-step equations:

This video shows how to solve a two step equation. It begins with the concept of equality, what is done to one side of an equation, must be done to the other side of an equation.

Type: Tutorial

Solving Multi-Step Equations:

This short video explains how to solve multi-step equations with variables on both sides and why it is necessary to complete the same steps on both sides of the equation.

Type: Tutorial

Solving Inconsistent or Dependent Systems:

When solving a system of linear equations in x and y with a single solution, we get a unique pair of values for x and y. But what happens when try to solve a system with no solutions or an infinite number of solutions?

Type: Tutorial

Linear Equations in One Variable:

This lesson introduces students to linear equations in one variable, shows how to solve them using addition, subtraction, multiplication, and division properties of equalities, and allows students to determine if a value is a solution, if there are infinitely many solutions, or no solution at all. The site contains an explanation of equations and linear equations, how to solve equations in general, and a strategy for solving linear equations. The lesson also explains contradiction (an equation with no solution) and identity (an equation with infinite solutions). There are five practice problems at the end for students to test their knowledge with links to answers and explanations of how those answers were found. Additional resources are also referenced.

Type: Tutorial

Solving Equations With the Variable on Both Sides.:

This video models solving equations in one variable with variables on both sides of the equal sign.

Type: Tutorial

Solving Equations with One Variable :

This Khan Academy presentation models solving two-step equations with one variable.

Type: Tutorial

## Video/Audio/Animation

Solving Mixture Problems with Linear Equations:

Mixture problems can involve mixtures of things other than liquids. This video shows how Algebra can be used to solve problems involving mixtures of different types of items.

Type: Video/Audio/Animation

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

Multi-Step Equations: Part 5 How Many Solutions?:

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• [CURRENT TUTORIAL] Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-Step Equations: Part 4 Putting it All Together:

Learn alternative methods of solving multi-step equations in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• [CURRENT TUTORIAL] Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-step Equations: Part 3 Variables on Both Sides:

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• [CURRENT TUTORIAL] Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-Step Equations: Part 2 Distributive Property:

Explore how to solve multi-step equations using the distributive property in this interactive tutorial.

This is part two of five in a series on solving multi-step equations.

• Click HERE to open Part 1: Combining Like Terms
• [CURRENT TUTORIAL] Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Multi-Step Equations: Part 1 Combining Like Terms:

Learn how to solve multi-step equations that contain like terms in this interactive tutorial.

This is part one of five in a series on solving multi-step equations.

• [CURRENT TUTORIAL] Part 1: Combining Like Terms
• Click HERE to open Part 2: The Distributive Property
• Click HERE to open Part 3: Variables on Both Sides
• Click HERE to open Part 4: Putting It All Together
• Click HERE to open Part 5: How Many Solutions?

Type: Original Student Tutorial

## Educational Games

Solving Equations: Same Variable, Both Sides, One Solution:

In this challenge game, you will be solving equations with variables on both sides. Each equation has a real solution. Use the "Teach Me" button to review content before the challenge. After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Educational Game

Interpreting the Graph:

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Quinoa Pasta 1:

This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns. This progression of tasks helps distinguish between 8th grade and high school expectations related to systems of linear equations.

Solving Equations:

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Cell Phone Plans:

This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the cell phone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task.

The Sign of Solutions:

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Two Lines:

In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.

Selling Fuel Oil at a Loss:

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table (Standard for Mathematical Practice, ).

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

Example 3: Solving Systems by Substitution:

This example demonstrates solving a system of equations algebraically and graphically.

Type: Tutorial

Substitution Method Example 2:

This video demonstrates a system of equations with no solution.

Type: Tutorial

The Substitution Method:

This video shows how to solve a system of equations using the substitution method.

Type: Tutorial

Checking Solutions to Systems of Equations Example:

This video demonstrates testing a solution (coordinate pair) for a system of equations

Type: Tutorial

Using a Graph to Analyze Solutions to Linear Systems:

This video demonstrates analyzing solutions to linear systems using a graph.

Type: Tutorial

Example of System with No Solution:

This video shows how to algebraically analyze a system that has no solutions.

Type: Tutorial

Solve a Consecutive Integer Problem Algebraically:

This video will show how to solve a consecutive integer problem.

Type: Tutorial

Distributive Property to Simplify Equations:

Use the Distributive Property while solving equations with variables on both sides.

Type: Tutorial

Introduction to solving an equation with variables on both sides:

Students will learn how to solve an equation with variables on both sides. This tutorial shows a final answer expressed as an improper fraction and mixed number.

Type: Tutorial

Application of the Distributive Property to Solve a Multi-Step Equation:

This video shows how to solve the equation (3/4)x + 2 = (3/8)x - 4 using the Distributive Property.

Type: Tutorial

Solving Equations with the Distributive Property:

This video shows how to solve an equation involving the Distributive Property.

Type: Tutorial

Solving a Multi-Step Equation:

This example involves a variable in the denominator on both sides of the equation.

Type: Tutorial

Solving an equation with variables on both sides:

Students will learn how to solve an equation with variables on both sides. Students will also learn how to distribute and combine like terms.

Type: Tutorial

Linear equation word problem:

Learn how to solve a word problem by writing an equation to model the situation. In this video, we use the linear equation 210(t-5) = 41,790.

Type: Tutorial

Solving Equations: Word Problem:

This tutorial shows a word problem in which students will find the dimensions of a garden given only the perimeter. Students will create an equation to solve.

Type: Tutorial

Solving a more complicated equation:

This example demonstrates how to solve an equation expressed in the form ax + b = c.

Type: Tutorial

Solving a two-step equation with a numerator of x:

This video shows how to solve an equation by isolating the variable in the numerator.

Type: Tutorial

Two-Step Equations:

Students will practice two step equations, some of which require combining like terms and using the distributive property.

Type: Tutorial

Solving two-step equations:

This video shows how to solve a two step equation. It begins with the concept of equality, what is done to one side of an equation, must be done to the other side of an equation.

Type: Tutorial

Solving Multi-Step Equations:

This short video explains how to solve multi-step equations with variables on both sides and why it is necessary to complete the same steps on both sides of the equation.

Type: Tutorial

Solving Inconsistent or Dependent Systems:

When solving a system of linear equations in x and y with a single solution, we get a unique pair of values for x and y. But what happens when try to solve a system with no solutions or an infinite number of solutions?

Type: Tutorial

Linear Equations in One Variable:

This lesson introduces students to linear equations in one variable, shows how to solve them using addition, subtraction, multiplication, and division properties of equalities, and allows students to determine if a value is a solution, if there are infinitely many solutions, or no solution at all. The site contains an explanation of equations and linear equations, how to solve equations in general, and a strategy for solving linear equations. The lesson also explains contradiction (an equation with no solution) and identity (an equation with infinite solutions). There are five practice problems at the end for students to test their knowledge with links to answers and explanations of how those answers were found. Additional resources are also referenced.

Type: Tutorial

Solving Equations With the Variable on Both Sides.:

This video models solving equations in one variable with variables on both sides of the equal sign.

Type: Tutorial

Solving Equations with One Variable :

This Khan Academy presentation models solving two-step equations with one variable.

Type: Tutorial

## Video/Audio/Animation

Solving Mixture Problems with Linear Equations:

Mixture problems can involve mixtures of things other than liquids. This video shows how Algebra can be used to solve problems involving mixtures of different types of items.

Type: Video/Audio/Animation

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Interpreting the Graph:

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Coupon Versus Discount:

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Quinoa Pasta 1:

This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns. This progression of tasks helps distinguish between 8th grade and high school expectations related to systems of linear equations.

Solving Equations:

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Cell Phone Plans:

This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the cell phone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task.

The Sign of Solutions:

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Two Lines:

In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.

Selling Fuel Oil at a Loss:

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table (Standard for Mathematical Practice, ).

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.