- Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Remarks
Fluency Expectations or Examples of Culminating StandardsStudents have been working informally with one-variable linear equations since as early as kindergarten. This important line of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers.
Examples of Opportunities for In-Depth Focus
This is a culminating standard for solving one-variable linear equations.
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.
- Assessment Limits :
Numbers in items must be rational numbers
- Calculator :
Yes
- Context :
Allowable
- Test Item #: Sample Item 1
- Question: How many solutions does the equation shown have?
- Difficulty: N/A
- Type: OR: Open Response
- Test Item #: Sample Item 2
- Question:
What values of a and b would make the equation shown have infinitely many solutions?
- Difficulty: N/A
- Type: EE: Equation Editor
- Test Item #: Sample Item 3
- Question:
Solve the equation shown for x.
- Difficulty: N/A
- Type: EE: Equation Editor
- Test Item #: Sample Item 4
- Question:
Explain why has no solution. Choose the best response.
- Difficulty: N/A
- Type: MC: Multiple Choice
- Test Item #: Sample Item 5
- Question:
Enter values of a and b for which x = 4 is a solution of the equation shown.
- Difficulty: N/A
- Type: EE: Equation Editor
- Test Item #: Sample Item 6
- Question:
Select whether each equation has no solution, one solution, or infinitely many solutions.
- Difficulty: N/A
- Type: MI: Matching Item
Related Courses
Related Access Points
Related Resources
Educational Games
Formative Assessments
Lesson Plans
Original Student Tutorials
Problem-Solving Tasks
Student Center Activity
Tutorials
Video/Audio/Animation
STEM Lessons - Model Eliciting Activity
This MEA provides students with opportunities to practice solving one-step equations while learning about density. Students will calculate density of regular and irregular objects.
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. Click here to learn more about MEAs and how they can transform your classroom.
MFAS Formative Assessments
Students are asked to identify and explain whether given equations have one solution, no solutions, or infinitely many solutions.
Students are asked to write examples of equations with one solution, no solutions, and infinitely many solutions.
Students are asked to solve a linear equation in one variable with rational coefficients.
Students are asked to solve a linear equation in one variable with rational coefficients.
Students are asked to solve a linear equation in one variable with rational coefficients and variables on both sides of the equation.
Original Student Tutorials Mathematics - Grades 6-8
Learn how to solve multi-step equations that contain like terms in this interactive tutorial.
This is part one of five in a series on solving multi-step equations.
- [CURRENT TUTORIAL] Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Explore how to solve multi-step equations using the distributive property in this interactive tutorial.
This is part two of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- [CURRENT TUTORIAL] Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial.
This is part five of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- [CURRENT TUTORIAL] Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Learn alternative methods of solving multi-step equations in this interactive tutorial.
This is part five of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- [CURRENT TUTORIAL] Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.
This is part five of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- [CURRENT TUTORIAL] Part 5: How Many Solutions?
Student Resources
Original Student Tutorials
Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.
This is part five of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- [CURRENT TUTORIAL] Part 5: How Many Solutions?
Type: Original Student Tutorial
Learn alternative methods of solving multi-step equations in this interactive tutorial.
This is part five of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- [CURRENT TUTORIAL] Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Type: Original Student Tutorial
Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial.
This is part five of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- [CURRENT TUTORIAL] Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Type: Original Student Tutorial
Explore how to solve multi-step equations using the distributive property in this interactive tutorial.
This is part two of five in a series on solving multi-step equations.
- Click HERE to open Part 1: Combining Like Terms
- [CURRENT TUTORIAL] Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Type: Original Student Tutorial
Learn how to solve multi-step equations that contain like terms in this interactive tutorial.
This is part one of five in a series on solving multi-step equations.
- [CURRENT TUTORIAL] Part 1: Combining Like Terms
- Click HERE to open Part 2: The Distributive Property
- Click HERE to open Part 3: Variables on Both Sides
- Click HERE to open Part 4: Putting It All Together
- Click HERE to open Part 5: How Many Solutions?
Type: Original Student Tutorial
Educational Games
In this challenge game, you will be solving equations with variables on both sides. Each equation has a real solution. Use the "Teach Me" button to review content before the challenge. After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!
Type: Educational Game
In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Educational Game
In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.
Type: Educational Game
Problem-Solving Tasks
In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.
Type: Problem-Solving Task
In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.
Type: Problem-Solving Task
It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.
Type: Problem-Solving Task
Student Center Activity
Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.
Type: Student Center Activity
Tutorials
This video will show how to solve a consecutive integer problem.
Type: Tutorial
Use the Distributive Property while solving equations with variables on both sides.
Type: Tutorial
Students will learn how to solve an equation with variables on both sides. This tutorial shows a final answer expressed as an improper fraction and mixed number.
Type: Tutorial
This video shows how to solve the equation (3/4)x + 2 = (3/8)x - 4 using the Distributive Property.
Type: Tutorial
This video shows how to solve an equation involving the Distributive Property.
Type: Tutorial
This example involves a variable in the denominator on both sides of the equation.
Type: Tutorial
Students will learn how to solve an equation with variables on both sides. Students will also learn how to distribute and combine like terms.
Type: Tutorial
Learn how to solve a word problem by writing an equation to model the situation. In this video, we use the linear equation 210(t-5) = 41,790.
Type: Tutorial
This tutorial shows a word problem in which students will find the dimensions of a garden given only the perimeter. Students will create an equation to solve.
Type: Tutorial
This example demonstrates how to solve an equation expressed in the form ax + b = c.
Type: Tutorial
This video shows how to solve an equation by isolating the variable in the numerator.
Type: Tutorial
Students will practice two step equations, some of which require combining like terms and using the distributive property.
Type: Tutorial
This video shows how to solve a two step equation. It begins with the concept of equality, what is done to one side of an equation, must be done to the other side of an equation.
Type: Tutorial
This short video explains how to solve multi-step equations with variables on both sides and why it is necessary to complete the same steps on both sides of the equation.
Type: Tutorial
This lesson introduces students to linear equations in one variable, shows how to solve them using addition, subtraction, multiplication, and division properties of equalities, and allows students to determine if a value is a solution, if there are infinitely many solutions, or no solution at all. The site contains an explanation of equations and linear equations, how to solve equations in general, and a strategy for solving linear equations. The lesson also explains contradiction (an equation with no solution) and identity (an equation with infinite solutions). There are five practice problems at the end for students to test their knowledge with links to answers and explanations of how those answers were found. Additional resources are also referenced.
Type: Tutorial
This video models solving equations in one variable with variables on both sides of the equal sign.
Type: Tutorial
This Khan Academy presentation models solving two-step equations with one variable.
Type: Tutorial
Video/Audio/Animation
Mixture problems can involve mixtures of things other than liquids. This video shows how Algebra can be used to solve problems involving mixtures of different types of items.
Type: Video/Audio/Animation
Parent Resources
Problem-Solving Tasks
In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.
Type: Problem-Solving Task
In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.
Type: Problem-Solving Task
It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.
Type: Problem-Solving Task
Tutorial
This video models solving equations in one variable with variables on both sides of the equal sign.
Type: Tutorial