Which Expression Is Equivalent To $35m^2 - 63$?

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Introduction

In algebra, equivalent expressions are those that have the same value for all possible values of the variable. In this article, we will explore the concept of equivalent expressions and find the expression that is equivalent to $35m^2 - 63$. We will use various mathematical techniques, including factoring and simplifying expressions, to arrive at the solution.

Understanding Equivalent Expressions

Equivalent expressions are expressions that have the same value for all possible values of the variable. In other words, if two expressions are equivalent, they will always produce the same result when evaluated. For example, the expressions $2x + 3$ and $x + 2 + 1$ are equivalent because they both represent the same value for all possible values of $x$.

Breaking Down the Given Expression

The given expression is $35m^2 - 63$. To find an equivalent expression, we need to analyze this expression and identify any patterns or structures that can be used to simplify it. Let's start by factoring out the greatest common factor (GCF) of the two terms.

Factoring Out the GCF

The GCF of $35m^2$ and $63$ is $7$. We can factor out $7$ from both terms to get:

$$7(5m^2) - 7(9)$$

This expression is equivalent to the original expression because we have simply factored out the GCF.

Simplifying the Expression

Now that we have factored out the GCF, we can simplify the expression further. We can rewrite the expression as:

$$7(5m^2 - 9)$$

This expression is equivalent to the original expression because we have simply factored out the GCF and rewritten the expression in a simpler form.

Using the Distributive Property

We can use the distributive property to expand the expression and simplify it further. The distributive property states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can use this property to expand the expression as follows:

$$7(5m^2 - 9) = 35m^2 - 63$$

This expression is equivalent to the original expression because we have simply expanded the expression using the distributive property.

Conclusion

In conclusion, the expression that is equivalent to $35m^2 - 63$ is $7(5m^2 - 9)$. We arrived at this solution by factoring out the GCF, simplifying the expression, and using the distributive property. This example illustrates the importance of using various mathematical techniques to simplify and manipulate expressions.

Final Answer

The final answer is: $\boxed{7(5m^2 - 9)}$

Additional Examples

Here are a few additional examples of equivalent expressions:

• $2x + 5$ and $x + 2 + 3$
• $3y - 2$ and $y - 1 - 1$
• $4z + 1$ and $z + 1 + 3$

These examples illustrate the concept of equivalent expressions and demonstrate how to find equivalent expressions using various mathematical techniques.

Tips and Tricks

Here are a few tips and tricks for working with equivalent expressions:

• Always look for the greatest common factor (GCF) when simplifying expressions.
• Use the distributive property to expand expressions and simplify them further.
• Be careful when simplifying expressions, as small mistakes can lead to incorrect solutions.

By following these tips and tricks, you can become proficient in working with equivalent expressions and solve a wide range of mathematical problems.

Conclusion

Introduction

In our previous article, we explored the concept of equivalent expressions and found the expression that is equivalent to $35m^2 - 63$. In this article, we will answer some frequently asked questions about equivalent expressions and provide additional examples and tips for working with them.

Q&A

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value for all possible values of the variable. In other words, if two expressions are equivalent, they will always produce the same result when evaluated.

Q: How do I find an equivalent expression?

A: To find an equivalent expression, you can use various mathematical techniques, including factoring and simplifying expressions, and using the distributive property.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. When simplifying expressions, it's often helpful to factor out the GCF.

Q: How do I use the distributive property?

A: The distributive property states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. You can use this property to expand expressions and simplify them further.

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

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

• Not factoring out the greatest common factor (GCF)
• Not using the distributive property to expand expressions
• Not simplifying expressions fully

Q: How do I know if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can substitute a value for the variable and evaluate both expressions. If they produce the same result, then they are equivalent.

Q: Can you provide some additional examples of equivalent expressions?

A: Here are a few additional examples of equivalent expressions:

• $2x + 5$ and $x + 2 + 3$
• $3y - 2$ and $y - 1 - 1$
• $4z + 1$ and $z + 1 + 3$

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

A: Equivalent expressions have many real-world applications, including:

• Simplifying algebraic expressions in physics and engineering
• Solving systems of linear equations in economics and finance
• Modeling population growth and decay in biology and medicine

Tips and Tricks

Here are a few additional tips and tricks for working with equivalent expressions:

• Always look for the greatest common factor (GCF) when simplifying expressions.
• Use the distributive property to expand expressions and simplify them further.
• Be careful when simplifying expressions, as small mistakes can lead to incorrect solutions.
• Practice, practice, practice! The more you work with equivalent expressions, the more comfortable you will become with them.

Conclusion

In conclusion, equivalent expressions are an important concept in algebra that can be used to simplify and manipulate expressions. By factoring out the GCF, simplifying expressions, and using the distributive property, we can find equivalent expressions and solve a wide range of mathematical problems. We hope this article has provided a clear and concise explanation of equivalent expressions and has helped you to become more proficient in working with them.

Additional Resources

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

• Khan Academy: https://www.khanacademy.org/math/algebra
• Mathway: https://www.mathway.com/
• Wolfram Alpha: https://www.wolframalpha.com/

We hope this article has been helpful in your understanding of equivalent expressions. If you have any further questions or need additional assistance, please don't hesitate to ask.