# Cluster 3: Solve systems of equations. (Algebra 1 - Additional Cluster) (Algebra 2 - Additional 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.912.A-REI.3
Title: Solve systems of equations. (Algebra 1 - Additional Cluster) (Algebra 2 - Additional Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Algebra: Reasoning with Equations & Inequalities

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Independent

MAFS.912.A-REI.3.In.7a
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically.
MAFS.912.A-REI.3.In.7b
Solve a simple system consisting of a linear equation and a quadratic equation in two variables graphically.

## Access Points

MAFS.912.A-REI.3.AP.5a
Create a multiple of a linear equation showing that they are equivalent (e.g., x + y = 6 is equivalent to 2x + 2y = 12).
MAFS.912.A-REI.3.AP.5b
Find the sum of two equations.
MAFS.912.A-REI.3.AP.6a
Given a graph, describe or select the solution to a system of linear equations.
MAFS.912.A-REI.3.AP.6b
Solve systems of nonlinear equations using substitution.

## Related Resources

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

## Formative Assessments

Solving Systems:

Students are given a system of two linear equations and asked to form a new system by replacing one equation with the sum of that equation and a multiple of the other. Then students are asked to explain why the two systems have the same solution.

Type: Formative Assessment

Solution Sets of Systems:

Students are asked to show that, given a system of two equations in two variables, replacing one equation with the sum of that equation and a multiple of another produces a system with the same solutions.

Type: Formative Assessment

Apples and Peaches:

Students are asked to solve a system of equations with rational solutions either algebraically or by graphing and are asked to justify the choice of method.

Type: Formative Assessment

Solving a System of Equations - 1:

Students are asked to solve a system of equations both algebraically and graphically.

Type: Formative Assessment

Solving a System of Equations - 3:

Students are asked to solve a system of equations both algebraically and graphically.

Type: Formative Assessment

Solving a System of Equations - 2:

Students are asked to solve a system of equations both algebraically and graphically.

Type: Formative Assessment

## Lesson Plans

Take Me Out to the Ball Game!:

Students will research ticket sales at five different stadiums. They will then select two stadiums and develop a word problem. They will then prove that there is one solution and write and explanation why. This lesson is a project-based task that students can use to show their understanding of solving systems of equations.

Type: Lesson Plan

Changes are Coming! But Does it Really Matter? Exploring the Effects of “Special” Manipulations on the Solutions in a System of Equations.:

A great introductory lesson for solving systems of equations algebraically. Use as a follow up lesson to solving systems of equations graphically. Students will explore graphs of systems to see how manipulating the equations affects the solutions (if at all).

Type: Lesson Plan

Space Equations:

In this lesson, students model the orbit of a satellite and the trajectory of a missile with a system of equations. They solve the equations both graphically and algebraically.

Type: Lesson Plan

Steel vs. Wooden Roller Coaster Lab:

This lesson is a Follow Up Activity to the Algebra Institute and allows students to apply their skills on analyzing bivariate data. This STEM lesson allows students the opportunity to investigate if there is a linear relationship between a coaster's height and speed. Using technology the students can determine the line of best fit, correlation coefficient and use the line for interpolation. This lesson also uses prior knowledge and has students solve systems of equations graphically to determine which type of coaster is faster.

Type: Lesson Plan

Manufacturing Designer Gear T-Shirts:

In this Algebra 1 MEA, students will solve systems of equations from data given in a table. Students will find the break-even point and rank the t-shirt manufacturing companies from best to worst.

Type: Lesson Plan

Solving Linear Equations in Two Variables:

This lesson unit is intended to help you assess how well students are able to formulate and solve problems using algebra and, in particular, to identify and help students who have the following difficulties solving a problem using two linear equations with two variables and interpreting the meaning of algebraic expressions.

Type: Lesson Plan

Optimization Problems: Boomerangs:

This lesson is designed to help students develop strategies for solving optimization problems. Such problems typically involve scenarios where limited resources must be used to greatest effect, as in, for example, the allocation of time and materials to maximize profit.

Type: Lesson Plan

Solving Systems of Equations by Substitution:

In this lesson, students will learn how to solve systems of equations using substitution. Students will have the opportunity for small group and whole class discussion related to using substitution.

Type: Lesson Plan

Systems of the Linear Round Table:

This lesson is a follow-up review of systems of linear equations. Students will complete a group activity called Simultaneous Round Table to solve given systems of equations. Students will solve by graphing, elimination, and substitution.  Each student will also perform error analysis on the work from their peers, which will allow them to help each other to correct those mistakes. Class will use data from error analysis to create a plan of action to decrease errors in their work.  Students will discuss the concepts and analyze problems with each other. These concepts were taught in an earlier lesson. This lesson will also help students identify common mistakes and find solutions to remedy them.

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

Graphing vs. Substitution. Which would you choose?:

Students will solve multiple systems of equations using two methods: graphing and substitution. This will help students to make a connection between the two methods and realize that they will indeed get the same solution graphically and algebraically.  Students will compare the two methods and think about ways to decide which method to use for a particular problem. This lesson connects prior instruction on solving systems of equations graphically with using algebraic methods to solve systems of equations.

Type: Lesson Plan

My Candles are MELTING!:

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph each to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Type: Lesson Plan

Exploring Systems with Piggies, Pizzas and Phones:

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

Type: Lesson Plan

When Two Lines Meet:

Learn how to graph a system of two equations in two variables, and find the solution(s), if one exists.

Type: Lesson Plan

## Original Student Tutorials

Solving Systems of Linear Equations Part 7: Word Problems:

Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 6: Writing Systems from Context:

Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.

This part 6 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing:

Learn to solve systems of linear equations by connecting algebraic and graphing methods in this interactive tutorial.

This part 5 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 4: Advanced Elimination:

Learn to solve systems of linear equations using advanced elimination in this interactive tutorial.

This part 4 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 3: Basic Elimination:

Learn to solve systems of linear equations using basic elimination in this interactive tutorial.

This part 3 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 2: Solving Systems of Linear Equations Part 2: Substitution

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)
Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 2: Substitution:

Learn to solve systems of linear equations using substitution in this interactive tutorial.

This part 2 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination (Coming soon)

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)

Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)

Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)

Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 1: Using Graphs:

Learn how to solve systems of linear equations graphically in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiasts

Solving Systems of Equations, Oceans & Climate:

Angela Dial discusses how she solves systems of equations to determine how the composition of ocean floor sediment has changed over 65 million years to help reveal more information regarding climate change.

Type: Perspectives Video: Professional/Enthusiast

Determining Strengths of Shark Models based on Scatterplots and Regression:

Chip Cotton, fishery biologist, discusses his use of mathematical regression modeling and how well the data fits his models based on  his deep sea shark research.

Type: Perspectives Video: Professional/Enthusiast

The Circle and The Line:

Although this task is fairly straightforward, it is worth noticing that it does not explicitly tell students to look for intersection points when they graph the circle and the line. Thus, in addition to assessing whether they can solve the system of equations, it is assessing a simple but important piece of conceptual understanding, namely the correspondence between intersection points of the two graphs and solutions of the system.

Quinoa Pasta 3:

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Quinoa Pasta 2:

This task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.

Pairs of Whole Numbers:

This task addresses A-REI.3.6, solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

This task asks students to consider the linear and quadratic functions shown on a graph, and use quadratic functions to find the coordinates.

Cash Box:

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not.

Accurately weighing pennies II:

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.

Accurately weighing pennies I:

This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.

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.

## Tutorials

Example 3: Solving Systems by Elimination:

This video is an example of solving a system of linear equations by elimination where the system has infinite solutions.

Type: Tutorial

Solving Systems of Linear Equations with Elimination Example 1:

This video shows how to solve a system of equations through simple elimination.

Type: Tutorial

Inconsistent Systems of Equations:

This video explains how to identify systems of equations without a solution.

Type: Tutorial

Example 2: Solving Systems by Elimination:

This video shows how to solve systems of equations by elimination.

Type: Tutorial

This video is an introduction to the elimination method of solving a system of equations.

Type: Tutorial

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

Systems of Equations Word Problems Example 1:

This video demonstrates solving a word problem by creating a system of linear equations that represents the situation and solving them using elimination.

Type: Tutorial

Graphing systems of equations:

In this tutorial, students will learn how to solve and graph a system of equations.

Type: Tutorial

Solving system of equations by graphing:

This tutorial shows students how to solve a system of linear equations by graphing the two equations on the same coordinate plane and identifying the intersection point.

Type: Tutorial

Solving a system of equations by graphing:

This tutorial shows how to solve a system of equations by graphing. Students will see what a no solution system of equations looks like in a graph.

Type: Tutorial

Solving a system of equations using substitution:

This tutorial shows how to solve a system of equations using substitution.

Type: Tutorial

Trolls, tolls, and systems of equations:

This video tutorial discusses how to create a system of equations.

Type: Tutorial

Solving Basic Systems Using the Elimination Method:

This 8 minute video will show step-by-step directions for using the elimination method to solve a system of linear equations.

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

Inconsistent, Dependent, and Independent Systems:

Systems of two linear equations in two variables can have a single solution, no solutions, or an infinite number of solutions. This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept.

Type: Tutorial

Solving Systems of Equations by Elimination:

Systems of two equations in x and y can be solved by adding the equations to create a new equation with one variable eliminated. This new equation can then be solved to find the value of the remaining variable. That value is then substituted into either equation to find the value of other variable.

Type: Tutorial

Solving Systems of Equations by Substitution:

A system of two equations in x and y can be solved by rearranging one equation to represent x in terms of y, and "substituting" this expression for x in the other equation. This creates an equation with only y which can then be solved to find y's value. This value can then be substituted into either equation to find the value of x.

Type: Tutorial

## Unit/Lesson Sequence

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

### Using this CMAP

To view an introduction on the CMAP tool, please .

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

Type: Unit/Lesson Sequence

## Video/Audio/Animations

Using Systems of Equations Versus One Equation:

When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?

Type: Video/Audio/Animation

Systems of Linear Equations in Two Variables:

The points of intersection of two graphs represent common solutions to both equations. Finding these intersection points is an important tool in analyzing physical and mathematical systems.

Type: Video/Audio/Animation

Why the Elimination Method Works:

This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.

Type: Video/Audio/Animation

Solving Systems by Substitution 1: No Solution:

This video demonstrates the substitution method to solve a system of equations, that there is no solution, and that the graphs of the equations are parallel.

Type: Video/Audio/Animation

## Student Resources

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

## Original Student Tutorials

Solving Systems of Linear Equations Part 7: Word Problems:

Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.

This part 7 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 6: Writing Systems from Context:

Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.

This part 6 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing:

Learn to solve systems of linear equations by connecting algebraic and graphing methods in this interactive tutorial.

This part 5 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 4: Advanced Elimination:

Learn to solve systems of linear equations using advanced elimination in this interactive tutorial.

This part 4 in a 7-part series. Click below to explore the other tutorials in the series.

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 3: Basic Elimination:

Learn to solve systems of linear equations using basic elimination in this interactive tutorial.

This part 3 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 2: Solving Systems of Linear Equations Part 2: Substitution

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)
Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 2: Substitution:

Learn to solve systems of linear equations using substitution in this interactive tutorial.

This part 2 in a 7-part series. Click below to explore the other tutorials in the series.

Part 1: Solving Systems of Linear Equations Part 1: Using Graphs

Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination (Coming soon)

Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)

Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)

Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)

Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 1: Using Graphs:

Learn how to solve systems of linear equations graphically in this interactive tutorial.

Type: Original Student Tutorial

The Circle and The Line:

Although this task is fairly straightforward, it is worth noticing that it does not explicitly tell students to look for intersection points when they graph the circle and the line. Thus, in addition to assessing whether they can solve the system of equations, it is assessing a simple but important piece of conceptual understanding, namely the correspondence between intersection points of the two graphs and solutions of the system.

Quinoa Pasta 3:

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Quinoa Pasta 2:

This task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.

Pairs of Whole Numbers:

This task addresses A-REI.3.6, solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

This task asks students to consider the linear and quadratic functions shown on a graph, and use quadratic functions to find the coordinates.

Cash Box:

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not.

Accurately weighing pennies II:

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.

Accurately weighing pennies I:

This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.

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.

## Tutorials

Example 3: Solving Systems by Elimination:

This video is an example of solving a system of linear equations by elimination where the system has infinite solutions.

Type: Tutorial

Solving Systems of Linear Equations with Elimination Example 1:

This video shows how to solve a system of equations through simple elimination.

Type: Tutorial

Inconsistent Systems of Equations:

This video explains how to identify systems of equations without a solution.

Type: Tutorial

Example 2: Solving Systems by Elimination:

This video shows how to solve systems of equations by elimination.

Type: Tutorial

This video is an introduction to the elimination method of solving a system of equations.

Type: Tutorial

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

Systems of Equations Word Problems Example 1:

This video demonstrates solving a word problem by creating a system of linear equations that represents the situation and solving them using elimination.

Type: Tutorial

Graphing systems of equations:

In this tutorial, students will learn how to solve and graph a system of equations.

Type: Tutorial

Solving system of equations by graphing:

This tutorial shows students how to solve a system of linear equations by graphing the two equations on the same coordinate plane and identifying the intersection point.

Type: Tutorial

Solving a system of equations by graphing:

This tutorial shows how to solve a system of equations by graphing. Students will see what a no solution system of equations looks like in a graph.

Type: Tutorial

Solving a system of equations using substitution:

This tutorial shows how to solve a system of equations using substitution.

Type: Tutorial

Trolls, tolls, and systems of equations:

This video tutorial discusses how to create a system of equations.

Type: Tutorial

Solving Basic Systems Using the Elimination Method:

This 8 minute video will show step-by-step directions for using the elimination method to solve a system of linear equations.

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

Inconsistent, Dependent, and Independent Systems:

Systems of two linear equations in two variables can have a single solution, no solutions, or an infinite number of solutions. This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept.

Type: Tutorial

Solving Systems of Equations by Elimination:

Systems of two equations in x and y can be solved by adding the equations to create a new equation with one variable eliminated. This new equation can then be solved to find the value of the remaining variable. That value is then substituted into either equation to find the value of other variable.

Type: Tutorial

Solving Systems of Equations by Substitution:

A system of two equations in x and y can be solved by rearranging one equation to represent x in terms of y, and "substituting" this expression for x in the other equation. This creates an equation with only y which can then be solved to find y's value. This value can then be substituted into either equation to find the value of x.

Type: Tutorial

## Video/Audio/Animations

Using Systems of Equations Versus One Equation:

When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?

Type: Video/Audio/Animation

Systems of Linear Equations in Two Variables:

The points of intersection of two graphs represent common solutions to both equations. Finding these intersection points is an important tool in analyzing physical and mathematical systems.

Type: Video/Audio/Animation

Why the Elimination Method Works:

This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.

Type: Video/Audio/Animation

## Parent Resources

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

The Circle and The Line:

Although this task is fairly straightforward, it is worth noticing that it does not explicitly tell students to look for intersection points when they graph the circle and the line. Thus, in addition to assessing whether they can solve the system of equations, it is assessing a simple but important piece of conceptual understanding, namely the correspondence between intersection points of the two graphs and solutions of the system.

Quinoa Pasta 3:

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Quinoa Pasta 2:

This task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.

Pairs of Whole Numbers:

This task addresses A-REI.3.6, solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

This task asks students to consider the linear and quadratic functions shown on a graph, and use quadratic functions to find the coordinates.

Cash Box:

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not.

Accurately weighing pennies II:

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.