# MA.912.AR.9.1 Export Print
Given a mathematical or real-world context, write and solve a system of two-variable linear equations algebraically or graphically.

### 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.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### 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.
• 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
• 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 subtract 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

• 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?

• 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

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200370: Algebra 1-A (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912080: Access Algebra 1A (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200375: Algebra 1-A for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
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.

## Related Resources

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

## Formative Assessments

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.

Type: Formative Assessment

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.

Type: Formative Assessment

System Solutions:

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

Type: Formative Assessment

Solving Systems of Linear Equations:

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

Type: Formative Assessment

Apples and Peaches:

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

Type: Formative Assessment

Solving a System of Equations - 1:

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

Type: Formative Assessment

Solving a System of Equations - 3:

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

Type: Formative Assessment

Solving a System of Equations - 2:

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

Type: Formative Assessment

## Lesson Plans

Take Me Out to the Ball Game!:

Students will research ticket sales at five different stadiums. They will then select two stadiums and develop a word problem. They will then 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.

Type: Lesson Plan

Solving Systems of Equations by Substitution:

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

Type: Lesson Plan

Systems of the Linear Round Table:

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

Type: Lesson Plan

Graphing vs. Substitution. Which would you choose?:

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

Type: Lesson Plan

My Candles are MELTING!:

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

Type: Lesson Plan

Exploring Systems with Piggies, Pizzas and Phones:

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

Type: Lesson Plan

## Original Student Tutorials

Solving Systems of Linear Equations Part 7: Word Problems:

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

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

Type: Original Student Tutorial

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

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

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

Type: Original Student Tutorial

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

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 4: Advanced Elimination:

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 3: Basic Elimination:

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

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

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

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 2: Substitution:

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

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

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

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

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

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

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 1: Using Graphs:

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

Type: Original Student Tutorial

Solving an Equation Using a Graph:

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

## Perspectives Video: Professional/Enthusiast

Determining Strengths of Shark Models based on Scatterplots and Regression:

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

Type: Perspectives Video: Professional/Enthusiast

## Tutorials

Example 3: Solving Systems by Elimination:

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

Type: Tutorial

Solving Systems of Linear Equations with Elimination Example 1:

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

Type: Tutorial

Inconsistent Systems of Equations:

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

Type: Tutorial

Example 2: Solving Systems by Elimination:

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

Type: Tutorial

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

Type: Tutorial

Systems of Equations Word Problems Example 1:

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

Type: Tutorial

Graphing systems of equations:

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

Type: Tutorial

Solving system of equations by graphing:

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

Type: Tutorial

Solving a system of equations by graphing:

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

Type: Tutorial

Solving a system of equations using substitution:

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

Type: Tutorial

Inconsistent, Dependent, and Independent Systems:

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

Type: Tutorial

Solving Systems of Equations by Elimination:

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

Type: Tutorial

Solving Systems of Equations by Substitution:

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

Type: Tutorial

## Video/Audio/Animations

Using Systems of Equations Versus One Equation:

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

Type: Video/Audio/Animation

Why the Elimination Method Works:

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

Type: Video/Audio/Animation

## MFAS Formative Assessments

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.

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.

Solving a System of Equations - 1:

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.

Solving a System of Equations - 3:

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

Solving Systems of Linear Equations:

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

System Solutions:

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

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.

## Original Student Tutorials Mathematics - Grades 9-12

Solving an Equation Using a Graph:

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.

Solving Systems of Linear Equations Part 1: Using Graphs:

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

Solving Systems of Linear Equations Part 2: Substitution:

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

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

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

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

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

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

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

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

Solving Systems of Linear Equations Part 3: Basic Elimination:

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

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

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

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

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

Solving Systems of Linear Equations Part 4: Advanced Elimination:

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

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

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

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

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

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

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

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

Solving Systems of Linear Equations Part 7: Word Problems:

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

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

## Student Resources

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

## Original Student Tutorials

Solving Systems of Linear Equations Part 7: Word Problems:

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

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

Type: Original Student Tutorial

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

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

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

Type: Original Student Tutorial

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

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 4: Advanced Elimination:

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 3: Basic Elimination:

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

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

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

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 2: Substitution:

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

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

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

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

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

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

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

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

Type: Original Student Tutorial

Solving Systems of Linear Equations Part 1: Using Graphs:

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

Type: Original Student Tutorial

Solving an Equation Using a Graph:

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

## Tutorials

Example 3: Solving Systems by Elimination:

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

Type: Tutorial

Solving Systems of Linear Equations with Elimination Example 1:

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

Type: Tutorial

Inconsistent Systems of Equations:

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

Type: Tutorial

Example 2: Solving Systems by Elimination:

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

Type: Tutorial

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

Type: Tutorial

Systems of Equations Word Problems Example 1:

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

Type: Tutorial

Graphing systems of equations:

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

Type: Tutorial

Solving system of equations by graphing:

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

Type: Tutorial

Solving a system of equations by graphing:

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

Type: Tutorial

Solving a system of equations using substitution:

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

Type: Tutorial

Inconsistent, Dependent, and Independent Systems:

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

Type: Tutorial

Solving Systems of Equations by Elimination:

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

Type: Tutorial

Solving Systems of Equations by Substitution:

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

Type: Tutorial

## Video/Audio/Animations

Using Systems of Equations Versus One Equation:

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

Type: Video/Audio/Animation

Why the Elimination Method Works:

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

Type: Video/Audio/Animation

## Parent Resources

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