Identify Equivalent Linear Expressions.Which Expression Is Equivalent To − 4 ( 5 R − 5 -4(5r - 5 − 4 ( 5 R − 5 ]?A. 20 R − 20 20r - 20 20 R − 20

Understanding Linear Expressions

Linear expressions are algebraic expressions that consist of a single term or the sum of several terms, each of which is a constant or a variable multiplied by a constant. In this article, we will focus on identifying equivalent linear expressions, which are expressions that have the same value for a given variable.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the constant outside the parentheses. The distributive property can be written as:

a(b + c) = ab + ac

where a, b, and c are constants or variables.

Applying the Distributive Property

To identify equivalent linear expressions, we need to apply the distributive property to the given expression. In this case, we have the expression:

$-4(5r - 5)$

Using the distributive property, we can expand this expression as follows:

$-4(5r - 5) = -4(5r) + (-4)(-5)$

$= -20r + 20$

Comparing the Expanded Expression

Now that we have expanded the given expression, we can compare it to the options provided. The first option is:

A. $20r - 20$

At first glance, this expression may seem equivalent to the expanded expression we obtained. However, let's take a closer look.

Analyzing the Options

To determine which expression is equivalent to the expanded expression, we need to analyze each option carefully.

Option A: $20r - 20$

This expression is not equivalent to the expanded expression we obtained. The sign of the term $20r$ is positive, whereas the sign of the term $-20r$ in the expanded expression is negative.

Option B: $-20r + 20$

This expression is equivalent to the expanded expression we obtained. The signs of the terms $-20r$ and $20$ are the same as the signs of the terms $-20r$ and $20$ in the expanded expression.

Option C: $20r + 20$

This expression is not equivalent to the expanded expression we obtained. The sign of the term $20r$ is positive, whereas the sign of the term $-20r$ in the expanded expression is negative.

Option D: $-20r - 20$

This expression is not equivalent to the expanded expression we obtained. The sign of the term $-20r$ is negative, whereas the sign of the term $-20r$ in the expanded expression is negative, but the sign of the term $20$ is positive.

Conclusion

In conclusion, the expression that is equivalent to $-4(5r - 5)$ is:

B. $-20r + 20$

This expression has the same value as the expanded expression we obtained using the distributive property.

Tips and Tricks

When working with linear expressions, it's essential to apply the distributive property to expand expressions and simplify them. By doing so, you can identify equivalent expressions and solve algebraic problems more efficiently.

Common Mistakes to Avoid

When working with linear expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Incorrect application of the distributive property: Make sure to apply the distributive property correctly to expand expressions.
• Incorrect simplification of expressions: Make sure to simplify expressions correctly by combining like terms.
• Incorrect comparison of expressions: Make sure to compare expressions carefully to identify equivalent expressions.

Practice Problems

To practice identifying equivalent linear expressions, try the following problems:

1. Identify the equivalent expression for $3(2x + 1)$.
2. Identify the equivalent expression for $-2(3x - 2)$.
3. Identify the equivalent expression for $4(2x + 3)$.

Answer Key

1. $6x + 3$
2. $-6x + 4$
3. $8x + 12$

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about identifying equivalent linear expressions.

Q: What is a linear expression?

A: A linear expression is an algebraic expression that consists of a single term or the sum of several terms, each of which is a constant or a variable multiplied by a constant.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the constant outside the parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term inside the parentheses by the constant outside the parentheses.

Q: What is the difference between equivalent and identical expressions?

A: Equivalent expressions are expressions that have the same value for a given variable, but may have different forms. Identical expressions are expressions that have the same form and value.

Q: How do I identify equivalent linear expressions?

A: To identify equivalent linear expressions, apply the distributive property to expand expressions and simplify them. Then, compare the expanded expression to the options provided to determine which expression is equivalent.

Q: What are some common mistakes to avoid when working with linear expressions?

A: Some common mistakes to avoid when working with linear expressions include:

• Incorrect application of the distributive property: Make sure to apply the distributive property correctly to expand expressions.
• Incorrect simplification of expressions: Make sure to simplify expressions correctly by combining like terms.
• Incorrect comparison of expressions: Make sure to compare expressions carefully to identify equivalent expressions.

Q: How can I practice identifying equivalent linear expressions?

A: You can practice identifying equivalent linear expressions by working through the practice problems provided in this article. Additionally, you can try creating your own practice problems by applying the distributive property to different expressions.

Q: What are some real-world applications of identifying equivalent linear expressions?

A: Identifying equivalent linear expressions has many real-world applications, including:

• Algebraic problem-solving: Identifying equivalent linear expressions is essential for solving algebraic problems, such as solving equations and inequalities.
• Data analysis: Identifying equivalent linear expressions is also important in data analysis, where it is used to model and analyze data.
• Science and engineering: Identifying equivalent linear expressions is used in science and engineering to model and analyze complex systems.

Conclusion

In conclusion, identifying equivalent linear expressions is a fundamental concept in algebra that has many real-world applications. By understanding the distributive property and applying it correctly, you can identify equivalent linear expressions and solve algebraic problems more efficiently. Remember to practice regularly and avoid common mistakes to become proficient in identifying equivalent linear expressions.

Additional Resources

For additional resources on identifying equivalent linear expressions, including videos, tutorials, and practice problems, visit the following websites:

• Khan Academy: www.khanacademy.org
• Mathway: www.mathway.com
• IXL: www.ixl.com

By following the steps outlined in this article and practicing regularly, you can become proficient in identifying equivalent linear expressions and solving algebraic problems more efficiently.