### Clarifications

*Clarification 1*: Within this benchmark, the expectation is to solve systems using elimination, substitution and graphing.

*Clarification 2*: Within the Algebra 1 course, the system is limited to two equations.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Algebraic Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Linear Equation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students determined whether a system of linear equations had one solution, no solution or infinitely many solutions and solved such systems graphically. In Algebra I, students solve systems of linear equations in two variables algebraically and graphically. In later courses, students will solve systems of linear equations in three variables and systems of nonlinear equations in two variables.- For students to have full understanding of systems, instruction should include MA.912.AR.9.4 and MA.912.AR.9.6. Equations and inequalities and their constraints are all related and the connections between them should be reinforced throughout instruction.
- Instruction allows students to solve using any method (substitution, elimination or
graphing) but recognizing that one method may be more efficient than another
*(MTR.3.1)*.- If both equations are given in standard form, then elimination, or linear combination, may be most efficient.
- If one equation is given in slope-intercept form or solved for $x$, then substitution may be easiest.
- If both equations are given in standard form, then elimination, or linear combination, may be most efficient.

- Consider presenting a system that favors one of these methods and having students divide
into three groups to solve them using different methods. Have students share their work
and discuss which method was more efficient than the others
*(*.*MTR.3.1*, MTR.4.1) - Include cases where students must interpret solutions to systems of equations.
- Instruction includes the use of various forms of linear equations.
- Standard Form

Can be described by the equation $A$$x$ + $B$$y$ = $c$, where $A$, $B$ and $C$ are any rational number. - Slope-Intercept Form

Can be described by the equation $y$ = $m$$x$ + $b$, where $m$ is the slope and $b$ is the $y$-intercept. - Point-Slope Form

Can be described by the equation $y$ − $y$_{1}= $m$($x$ − $x$_{1}), where ($x$_{1}, $y$_{1}) are a point on the line and $m$ is the slope of the line.

- Standard Form
- When introducing the elimination method, students may express confusion when considering adding equations together. Historically, students have used the properties of equality to create equivalent equations to solve for a variable of interest. In most of these efforts, operations performed on both sides of the original equation have been identical. With the introduction of the elimination method, students can now see that operations performed on each side of an equation must be equivalent (not necessarily identical) for the property to hold. Guide students to explore forming equivalent equations with simpler equations by adding or subtracting equivalent values. Lead them to see that the new equations they generate have the same solutions. Have them discuss why the method works: equations are simply pairs of equivalent expressions, which is why they can be added/subtracted with each other.

### Common Misconceptions or Errors

- Students may not understand linear systems of equations can only have more than one solution if there are infinitely many solutions.
- Students may not understand linear systems of equations can have no solution.
- Students may have difficulty making connections between graphic and algebraic representations of systems of equations.
- Students may have difficulty choosing the best method of finding the solution to a system of equations.
- Students may have difficulty translating word problems into systems of equations and inequalities.
- Students using the elimination method may alter the original equations in a way that creates like terms that can be subtracted. When subtracting across the two equations students may have difficulty remembering to apply the subtraction to the remaining terms and constants.

### Strategies to Support Tiered Instruction

- Instruction includes opportunities to use graphing software to visualize the possible solutions for a system of equations. Systems of equations only produce three different types of solutions: one solution, infinite solutions, and no solutions. Each type of system can be graphed for analysis of each type of solution set.
- Teacher models through a think-aloud how a system of equations can have no solutions.
- For example, “I can algebraically solve a system with no solutions. The solution will reveal that the left and right sides of the equation cannot be equal, causing a no solution set. In addition, if I rearrange both equations to the slope-intercept form, the equations will have the same slope. I can utilize my knowledge of parallel lines to understand that the system cannot have any solutions.”

- Teacher provides step-by-step process for solving systems.
- For example, when solving the system below, students can use the method of
elimination.2$x$ + 4$y$ = −10

3$x$ + 5$y$ = 8

- For example, when solving the system below, students can use the method of
elimination.

- If the student chooses to eliminate the $y$-variable, they can multiply the first
equation by 5 and the second by 4 so that both coefficients of $y$ are 20. 5(2$x$ + 4$y$ = −10) to 10$x$ + 20$y$ = −50

4(3$x$ + 5$y$ = 8) to 12$x$ + 20$y$ = 32

- The student either subtracts the two new equations or creates additive inverses by
multiplying one of the equations by −1 (as shown) and then adds the equations. −1(10$x$ + 20$y$ = −50) to −10$x$ − 20$y$ = +50

−10$x$ − 20$y$ = +50__12$x$ + 20$y$ = 32__

2$x$ = 82

$x$ = 41

- Once students determine one of the values ($x$ in this case), then they can
substitute this back into one of the given equations to find the other value ($y$ in
this case). 2(41) + 4$y$ = −10

4$y$ = −10−82

$y$ = −23

### Instructional Tasks

*Instructional Task 1 (*

*MTR.3.1*,*MTR.4.1*)- You and a friend go to Tacos Galore for lunch. You order three soft tacos and three burritos,
and your total bill is $11.25. Your friend’s bill is $10.00 for four soft tacos and two burritos.
- Part A. Write a system of two-variable linear equations to represent this situation.
- Part B. Solve the system both algebraically and graphically to determine the cost of each burrito and each soft taco.
- Part C. Is one method more efficient than the other? Why or why not?

Instructional Task 2 (

Instructional Task 2 (

*MTR.3.1*,*MTR.4.1*)- Part A. Determine the solution to the system of linear equations below using your method of
choice. 0.5$x$ − 1.4$y$ = 5.8

$y$ = −0.3$x$ − $\frac{\text{1}}{\text{5}}$

- Part B. Discuss with a partner why you chose that method.

### Instructional Items

*Instructional Item 1*

- Determine the exact solution of the system of linear equations below.−$\frac{\text{1}}{\text{10}}$$x$ + $\frac{\text{1}}{\text{2}}$$y$ = $\frac{\text{4}}{\text{5}}$

$\frac{\text{1}}{\text{7}}$$x$ + $\frac{\text{1}}{\text{3}}$$y$ = −$\frac{\text{2}}{\text{21}}$

*Instructional Item 2*

- Carla volunteered to make pies for a bake sale. She bought two pounds of apples and six pounds of peaches and spent $19. After baking the pies, she decided they looked so good she would make more. She went back to the store and bought another pound of apples and five more pounds of peaches and spent $15. Write a system of linear equations that describes her purchases, where $a$ represents the cost per pound of the apples and $p$ represents the cost per pound of the peaches.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

## Tutorials

## Video/Audio/Animations

## STEM Lessons - Model Eliciting Activity

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

## 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 determine the number of solutions of each of four systems of linear equations without solving the systems of equations.

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.

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

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

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

## Original Student Tutorials Mathematics - Grades 9-12

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.

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 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)

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)

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)

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)

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)

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

## 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 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 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

## Problem-Solving Tasks

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving 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 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.

Type: Problem-Solving 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 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.

Type: Problem-Solving 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: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving 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 short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task

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.

Type: Problem-Solving 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 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

## 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

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

## Problem-Solving Tasks

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving 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 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.

Type: Problem-Solving 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 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.

Type: Problem-Solving 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: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving 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 short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task