Standard 9: Write and solve a system of two- and three-variable equations and inequalities that describe quantities or relationships.

Students are asked to graph a step function, state the domain of the function, and name any intercepts.

Type: Formative Assessment

Students are asked to represent constraints using inequalities given in a problem context.

Type: Formative Assessment

Students are asked to model a problem involving constraints using inequalities.

Type: Formative Assessment

Students are asked to use technology (e.g., spreadsheet, graphing calculator, or dynamic geometry software) to estimate the solutions of the equation f(x) = g(x) for given functions f and g.

Type: Formative Assessment

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

Students are asked to find the solution(s) of the equation f(x) = g(x) given the graphs of f and g and explain their reasoning.

Type: Formative Assessment

Students are asked to find solutions of the equation f(x) = g(x) for two given functions, f and g, by constructing a table of values.

Type: Formative Assessment

Students are asked to graph a system of two linear inequalities.

Type: Formative Assessment

Students are asked to select the correct graph of the solution region of a given system of two linear inequalities.

Type: Formative Assessment

Students are asked to explain why the x-coordinate of the intersection of two functions, f and g, is a solution of the equation f(x) = g(x).

Type: Formative Assessment

Students are given a graph and a verbal description of a function and are asked to describe its domain.

Type: Formative Assessment

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

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

Type: Formative Assessment

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

Type: Formative Assessment

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

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

Students use systems of equations and inequalities to solve real world budgeting problems involving two variables.

Type: Lesson Plan

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

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

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

Type: Lesson Plan

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.

• Part 1: Solving Systems of Linear Equations: Using Graphs
• Part 2: Solving Systems of Linear Equations: Substitution
• Part 3: Solving Systems of Linear Equations: Basic Elimination
• Part 4: Solving Systems of Linear Equations: Advanced Elimination
• Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
• Part 6: Solving Systems of Linear Equations: Writing Systems from Context

Type: Original Student Tutorial

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. 

• Part 1: Solving Systems of Linear Equations: Using Graphs
• Part 2: Solving Systems of Linear Equations: Substitution
• Part 3: Solving Systems of Linear Equations: Basic Elimination
• Part 4: Solving Systems of Linear Equations: Advanced Elimination
• Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
• Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)

Type: Original Student Tutorial

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. 

• Part 1: Solving Systems of Linear Equations: Using Graphs
• Part 2: Solving Systems of Linear Equations: Substitution
• Part 3: Solving Systems of Linear Equations: Basic Elimination
• Part 4: Solving Systems of Linear Equations: Advanced Elimination
• Part 6: Solving Systems of Linear Equations: Writing Systems from Context (Coming soon)
• Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)

Type: Original Student Tutorial

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.

• Part 1: Solving Systems of Linear Equations Part 1: Using Graphs
• Part 2: Solving Systems of Linear Equations Part 2: Substitution
• Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination
• 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

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

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

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

Type: Original Student Tutorial

Follow as we learn why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x) in this interactive tutorial.

Type: Original Student Tutorial

In this video, Brad Rosenheim describes how Louisiana sediment cores are used to estimate sea level changes over the last 10,000 years. Video funded by NSF grant #: OCE-1502753.

Type: Perspectives Video: Expert

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

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.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

What happens when math models go wrong in forecasting hurricanes?

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Have a need for speed? Get out your spreadsheet! Race car drivers use algebraic formulas and spreadsheets to optimize car performance.

Type: Perspectives Video: Professional/Enthusiast

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

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

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

Type: Tutorial

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

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

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

Type: Tutorial

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

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

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

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

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