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

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MA.912.AR.9.1 : Given a mathematical or real-world context, write and solve a system of two-variable linear equations algebraically or graphically.
  • MA.912.AR.9.2 : Given a mathematical or real-world context, solve a system consisting of a two-variable linear equation and a non-linear equation algebraically or graphically.
  • MA.912.AR.9.3 : Given a mathematical or real-world context, solve a system consisting of two-variable linear or non-linear equations algebraically or graphically.
  • MA.912.AR.9.4 : Graph the solution set of a system of two-variable linear inequalities.
  • MA.912.AR.9.5 : Graph the solution set of a system of two-variable inequalities.
  • MA.912.AR.9.6 : Given a real-world context, represent constraints as systems of linear equations or inequalities. Interpret solutions to problems as viable or non-viable options.
  • MA.912.AR.9.7 : Given a real-world context, represent constraints as systems of linear and non-linear equations or inequalities. Interpret solutions to problems as viable or non-viable options.
  • MA.912.AR.9.8 : Solve real-world problems involving linear programming in two variables.
  • MA.912.AR.9.9 : Given a mathematical or real-world context, solve a system of three-variable linear equations algebraically.
  • MA.912.AR.9.10 : Solve and graph mathematical and real-world problems that are modeled with piecewise functions. Interpret key features and determine constraints in terms of the context.
Cluster Information
Number:
MA.912.AR.9
Title:
Write and solve a system of two- and three-variable equations and inequalities that describe quantities or relationships.
Type:
Standard
Subject:
Mathematics (B.E.S.T.)
Grade:
912
Strand
Algebraic Reasoning
Cluster Access Points

This cluster includes the following Access Points.

  • MA.912.AR.9.AP.1 : Given an algebraic or graphical system of two-variable linear equations, select the solution to the system of equations.
  • MA.912.AR.9.AP.2 : Solve a system consisting of a two-variable linear equation and a quadratic equation algebraically or graphically.
  • MA.912.AR.9.AP.3 : Solve a system consisting of two-variable linear or quadratic equations algebraically or graphically.
  • MA.912.AR.9.AP.4 : Select the graph of the solution set of a system of two-variable linear inequalities.
  • MA.912.AR.9.AP.5 : Select the graph of the solution set of a system of two-variable inequalities.
  • MA.912.AR.9.AP.6 : Given a real-world context, as systems of linear equations or inequalities with identified constraints, select a solution as a viable or non-viable option.
  • MA.912.AR.9.AP.7 : Given a real-world context, as systems of linear and non-linear equations or inequalities with identified constraints, select a solution as a viable or non-viable option.
Cluster Resources

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

Original Student Tutorials
Formative Assessments
  • Graphing a Step Function: Students are asked to graph a step function, state the domain of the function, and name any intercepts.

  • The New School: Students are asked to represent constraints using inequalities given in a problem context.

  • Sugar and Protein: Students are asked to model a problem involving constraints using inequalities.

  • Using Technology: 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.

  • 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.

  • Graphs and Solutions - 2: 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.

  • Using Tables: 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.

  • Graph a System of Inequalities: Students are asked to graph a system of two linear inequalities.

  • Which Graph?: Students are asked to select the correct graph of the solution region of a given system of two linear inequalities.

  • Graphs and Solutions -1: 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).

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

  • 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.

  • System Solutions: Students are asked to solve a word problem by solving a system of linear equations.

  • Solving Systems of Linear Equations: Students are asked to solve three systems of linear equations algebraically.

  • 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.

  • Solving a System of Equations - 1: Students are asked to solve a system of equations both algebraically and graphically.

  • Solving a System of Equations - 3: Students are asked to solve a system of equations both algebraically and graphically.

  • Solving a System of Equations - 2: Students are asked to solve a system of equations both algebraically and graphically.

Lesson Plans
  • How Shady Are You?: This lesson allows students to graph systems of linear inequalities with two variables and view three possible types of solutions.

  • 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 solve to show 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.

  • Changes are Coming to System of Equations: 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).

  • 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.

  • Solving Systems of Inequalities: Students will learn to graph a system of inequalities and identify points in the solution set. This lesson aligns with the Mathematics Formative Assessment System (MFAS) Task Graph a System of Inequalities (CPALMS Resource #60567). In this lesson, students with similar instructional needs are grouped according to MFAS rubric levels: Getting Started, Moving Forward, Almost There, and Got It. Students in each group complete an exercise designed to move them toward a better understanding of solutions of systems of inequalities and their graphs.

  • My first credit card!: In this Model Eliciting Activity, MEA, students use compare and analyze various features of credit cards to choose the best one for a college student. As part of their analysis, students will create step functions to model the interest charged and visually compare interest costs associated with each credit card.

    Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

  • Manufacturing Designer Gear T-Shirts: In this Model Eliciting Activity, MEA, students will analyze data, write and graph systems of equations, and develop a procedure to rank and justify manufacturing companies to produce t-shirts. Students will also use their functions to determine the break-even point given each manufacturer.

    Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

  • Solving Linear Equations in Two Variables: This lesson unit is intended to help you assess how well students can formulate and solve problems using algebra and, in particular, to identify and help students who have difficulties solving a problem using two linear equations with two variables and interpreting the meaning of algebraic expressions.

  • A Scheme for Solving Systems: Students will graph systems of linear equations in slope-intercept form to find the solution to the system. Students will practice with systems that have one solution, no solution, and all solutions. 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.

  • 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.

  • Don't Blow the Budget!: Students use systems of equations and inequalities to solve real world budgeting problems involving two variables.

  • 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.

  • 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.

  • 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.

  • Exploring Systems with Piggies, Pizzas and Phones: Students write and solve linear equations from real-life situations. 

Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Perspectives Video: Teaching Idea
  • Absolute Value Progression: Unlock an effective teaching strategy for making connections with absolute values to graphing in this Teacher Perspectives video for educators.

Problem-Solving Tasks
  • Centerpiece: The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).

  • 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.

  • Solution Sets: The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?

  • Quinoa Pasta 3: This mathematical modeling task also illustrates 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 mathematical modeling task also illustrates 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 solving systems of linear equations, 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.

  • 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 the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether 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.

  • 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.

  • 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.

  • Introduction to Linear Functions: This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

  • 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.

  • 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.

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.

  • Solving Systems of Linear Equations with Elimination Example 1: This video shows how to solve a system of equations through simple elimination.

  • Inconsistent Systems of Equations: This video explains how to identify systems of equations without a solution.

  • Example 2: Solving Systems by Elimination: This video shows how to solve systems of equations by elimination.

  • Addition Elimination Example 1: This video is an introduction to the elimination method of solving a system of equations.

  • 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.

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

     

  • 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. 

     

  • 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.

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

     

  • 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.

  • 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.

  • 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.

Video/Audio/Animations