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.

Assessed with:
MAFS.912.ACED.12
 Test Item #: Sample Item 1
 Question:
A system of equations is shown.
5x+2y=24
5y=x+6
Click on each blank to enter a numeric value to complete the solution to the system.
 Difficulty: N/A
 Type: EE: Equation Editor
 Test Item #: Sample Item 2
 Question:
Magda and Nicholas each purchase almonds and cashews for their trail mix recipes as shown,
Click on each blank to enter the cost per pound of almonds and cashews.
 Difficulty: N/A
 Type: EE: Equation Editor
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Professional/Enthusiast
ProblemSolving Tasks
Tutorials
Unit/Lesson Sequence
Video/Audio/Animations
STEM Lessons  Model Eliciting Activity
In this Algebra 1 MEA, students will solve systems of equations from data given in a table. Students will find the breakeven point and rank the tshirt manufacturing companies from best to worst.
Model Eliciting Activities, MEAs, are openended, interdisciplinary problemsolving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
MFAS Formative Assessments
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.
Students are asked to solve a system of equations both algebraically and graphically.
Students are asked to solve a system of equations both algebraically and graphically.
Students are asked to solve a system of equations both algebraically and graphically.
Original Student Tutorials Mathematics  Grades 912
Learn how to solve systems of linear equations graphically in this interactive tutorial.
Learn to solve systems of linear equations using substitution in this interactive tutorial.
This part 2 in a 7part 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)
Learn to solve systems of linear equations using basic elimination in this interactive tutorial.
This part 3 in a 7part 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)
Learn to solve systems of linear equations using advanced elimination in this interactive tutorial.
This part 4 in a 7part 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)
Learn to solve systems of linear equations by connecting algebraic and graphing methods in this interactive tutorial.
This part 5 in a 7part 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)
Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.
This part 6 in a 7part 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)
Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.
This part 7 in a 7part 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
Student Resources
Original Student Tutorials
Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.
This part 7 in a 7part 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 7part 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 7part 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 7part 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 7part 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 7part 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
ProblemSolving Tasks
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
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
This task addresses AREI.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.
Type: ProblemSolving Task
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 nonnegative 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.
Type: ProblemSolving Task
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 problemsolving (numerical reasoning, informed guessandcheck, 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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising shortterm capital for investment or reinvestment in developing the business.
Type: ProblemSolving Task
Tutorials
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 example demonstrates solving a system of equations algebraically and graphically.
Type: Tutorial
This video demonstrates a system of equations with no solution.
Type: Tutorial
This video shows how to solve a system of equations using the substitution method.
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
This 8 minute video will show stepbystep directions for using the elimination method to solve a system of linear equations.
Type: Tutorial
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
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
Video/Audio/Animations
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
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
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
ProblemSolving Tasks
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
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
This task addresses AREI.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.
Type: ProblemSolving Task
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 nonnegative 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.
Type: ProblemSolving Task
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 problemsolving (numerical reasoning, informed guessandcheck, 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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising shortterm capital for investment or reinvestment in developing the business.
Type: ProblemSolving Task