Mrs. McAlister Wrote The Equation ${ 10t - 4t + 3t - 8\$} On The Board And Asked Students To Write An Equivalent Equation. The Table Below Shows The Responses From Four

Mar 9, 2025 by ADMIN 169 views

**Simplifying Linear Equations: A Step-by-Step Guide**

Introduction

In mathematics, simplifying linear equations is an essential skill that helps students understand the underlying structure of algebraic expressions. When Mrs. McAlister asked her students to write an equivalent equation, she was testing their ability to simplify linear equations. In this article, we will explore the concept of simplifying linear equations, provide a step-by-step guide, and analyze the responses from four students.

What are Linear Equations?

A linear equation is an algebraic expression that can be written in the form of ax + b = c, where a, b, and c are constants, and x is the variable. Linear equations can be represented graphically as a straight line on a coordinate plane. In the given equation ${ $10t - 4t + 3t - 8\$}$ , we have a linear equation with a variable t.

Simplifying Linear Equations

To simplify a linear equation, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In the given equation, we have three like terms: 10t, -4t, and 3t. We can combine these terms by adding or subtracting their coefficients.

Step-by-Step Guide to Simplifying Linear Equations

Identify like terms: The first step is to identify like terms in the equation. In the given equation, we have three like terms: 10t, -4t, and 3t.
Combine like terms: Once we have identified like terms, we can combine them by adding or subtracting their coefficients. In this case, we can combine 10t, -4t, and 3t by adding their coefficients: 10t - 4t + 3t = (10 - 4 + 3)t = 9t.
Simplify the equation: After combining like terms, we can simplify the equation by rewriting it in a more compact form. In this case, we can rewrite the equation as 9t - 8.

Analyzing Student Responses

Let's analyze the responses from four students:

Student 1

Student 1 wrote the equation ${ $10t - 4t + 3t - 8\$}$ as ${ $9t - 8\$}$ . This student correctly combined like terms and simplified the equation.

Student 2

Student 2 wrote the equation ${ $10t - 4t + 3t - 8\$}$ as ${ $13t - 8\$}$ . This student incorrectly combined like terms and added the coefficients instead of subtracting them.

Student 3

Student 3 wrote the equation ${ $10t - 4t + 3t - 8\$}$ as ${ $10t - 4t + 3t + 8\$}$ . This student incorrectly added the constant term instead of subtracting it.

Student 4

Student 4 wrote the equation ${ $10t - 4t + 3t - 8\$}$ as ${ $10t - 4t - 3t - 8\$}$ . This student incorrectly combined like terms and subtracted the coefficients instead of adding them.

Conclusion

Simplifying linear equations is an essential skill in mathematics that helps students understand the underlying structure of algebraic expressions. By following a step-by-step guide, students can simplify linear equations and write equivalent expressions. In this article, we analyzed the responses from four students and identified common mistakes that students make when simplifying linear equations. By understanding these mistakes, teachers can provide targeted support to help students improve their skills.

Common Mistakes to Avoid

When simplifying linear equations, students often make the following mistakes:

Incorrectly combining like terms: Students may add or subtract coefficients instead of combining like terms.
Incorrectly adding or subtracting constant terms: Students may add or subtract constant terms instead of combining like terms.
Not simplifying the equation: Students may not simplify the equation after combining like terms.

Tips for Teachers

When teaching students to simplify linear equations, teachers can use the following tips:

Provide clear instructions: Teachers should provide clear instructions on how to simplify linear equations.
Use visual aids: Teachers can use visual aids such as graphs or charts to help students understand the concept of simplifying linear equations.
Provide practice exercises: Teachers can provide practice exercises to help students develop their skills in simplifying linear equations.

Conclusion

Introduction

Simplifying linear equations is a fundamental concept in mathematics that helps students understand the underlying structure of algebraic expressions. In our previous article, we explored the concept of simplifying linear equations and provided a step-by-step guide. In this article, we will answer frequently asked questions about simplifying linear equations.

Q: What are like terms in a linear equation?

A: Like terms are terms that have the same variable raised to the same power. In the equation ${ $10t - 4t + 3t - 8\$}$ , 10t, -4t, and 3t are like terms because they all have the variable t raised to the power of 1.

Q: How do I combine like terms in a linear equation?

A: To combine like terms, you need to add or subtract their coefficients. In the equation ${ $10t - 4t + 3t - 8\$}$ , you can combine 10t, -4t, and 3t by adding their coefficients: 10t - 4t + 3t = (10 - 4 + 3)t = 9t.

Q: What is the difference between adding and subtracting coefficients?

A: When adding coefficients, you are combining the terms by adding their values. When subtracting coefficients, you are combining the terms by subtracting their values. For example, in the equation ${ $10t - 4t + 3t - 8\$}$ , you can combine 10t, -4t, and 3t by adding their coefficients: 10t - 4t + 3t = (10 - 4 + 3)t = 9t. If you were to subtract the coefficients, you would get: 10t - 4t - 3t = (10 - 4 - 3)t = 3t.

Q: Can I simplify a linear equation with multiple variables?

A: Yes, you can simplify a linear equation with multiple variables. However, you need to identify like terms and combine them separately for each variable. For example, in the equation ${ $2x + 3y - 4x + 2y - 8\$}$ , you can combine like terms for x and y separately: (2x - 4x) + (3y + 2y) = -2x + 5y - 8.

Q: How do I know if I have simplified a linear equation correctly?

A: To check if you have simplified a linear equation correctly, you can:

Check if you have combined like terms: Make sure you have combined all like terms in the equation.
Check if the equation is in a simpler form: Make sure the equation is in a simpler form, such as a single term or a combination of terms with a common variable.
Check if the equation is equivalent to the original equation: Make sure the simplified equation is equivalent to the original equation.

Q: What are some common mistakes to avoid when simplifying linear equations?

A: Some common mistakes to avoid when simplifying linear equations include:

Incorrectly combining like terms: Adding or subtracting coefficients instead of combining like terms.
Incorrectly adding or subtracting constant terms: Adding or subtracting constant terms instead of combining like terms.
Not simplifying the equation: Not simplifying the equation after combining like terms.