Understanding the Concept of Equivalent Expressions

Oct 13, 2025 by miodeya.com 75 views

Introduction

In algebra, equivalent expressions are mathematical expressions that have the same value, even though they may look different. Understanding equivalent expressions is crucial in solving algebraic equations and simplifying complex expressions. In this article, we will explore two expressions that are equivalent to ${$ (-7)(-15)(-5)$}$ and examine the options provided to identify the correct pair of equivalent expressions.

Understanding the Concept of Equivalent Expressions

Equivalent expressions are mathematical expressions that have the same value, but may differ in their appearance. They can be obtained by applying various mathematical operations, such as addition, subtraction, multiplication, and division, to the original expression. For example, the expressions ${ $2x + 3\$}$ and ${ $3 + 2x\$}$ are equivalent because they have the same value, even though they are written differently.

Properties of Multiplication

When multiplying numbers, we can apply the commutative and associative properties to simplify the expression. The commutative property states that the order of the numbers being multiplied does not change the result, while the associative property states that the order in which we multiply the numbers does not change the result. For example, ${$ (3)(4)(5)$] is equivalent to [ $(3)(5)(4)\$] and also equivalent to \[$ (4)(3)(5)$].

Applying the Properties to the Given Expression

Now, let's apply the properties of multiplication to the given expression [ $(-7)(-15)(-5)\$]. We can start by multiplying the first two numbers, \[$ (-7)(-15)$], which gives us [ $105\$. Then, we multiply the result by the third number, \[$ (-5)$, which gives us [$(-525)$].

Examining the Options

Now that we have simplified the given expression, let's examine the options provided to identify the correct pair of equivalent expressions.

Option A: [ $(-7)(-75)\$] and \[$ (-1)(525)$]

To determine if these expressions are equivalent, we can simplify them separately. The first expression, [$(-7)(-75)$, is equivalent to [ $525\$. The second expression, \[$ (-1)(525)$, is equivalent to [$(-525)$. Since these expressions have different values, they are not equivalent.

Option B: [ $(-1)(525)\$] and \[$ (35)(-15)$]

To determine if these expressions are equivalent, we can simplify them separately. The first expression, [ $(-1)(525)\$, is equivalent to \[$ (-525)$. The second expression, [ $(35)(-15)\$, is equivalent to \[$ (-525)$. Since these expressions have the same value, they are equivalent.

Option C: [ $(35)(-15)\$] and \[$ (115)(-5)$]

To determine if these expressions are equivalent, we can simplify them separately. The first expression, [ $(35)(-15)\$, is equivalent to \[$ (-525)$. The second expression, [ $(115)(-5)\$, is equivalent to \[$ (-575)$. Since these expressions have different values, they are not equivalent.

Conclusion

In conclusion, the correct pair of equivalent expressions to [ $(-7)(-15)(-5)\$] is \[$ (-1)(525)$] and [$(35)(-15)$. These expressions have the same value, even though they may look different. Understanding equivalent expressions is crucial in solving algebraic equations and simplifying complex expressions.

Final Answer

$The final answer is B. \[$(-1)(525)\$] and \[$(35)(-15)\$.<br/> **Frequently Asked Questions (FAQs) About Equivalent Expressions**$

Introduction

In our previous article, we explored the concept of equivalent expressions in algebra and examined two expressions that are equivalent to [$(-7)(-15)(-5)$. In this article, we will answer some frequently asked questions (FAQs) about equivalent expressions to help you better understand this concept.

Q: What are equivalent expressions?

A: Equivalent expressions are mathematical expressions that have the same value, even though they may look different. They can be obtained by applying various mathematical operations, such as addition, subtraction, multiplication, and division, to the original expression.

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

A: To determine if two expressions are equivalent, you can simplify them separately and compare their values. If they have the same value, then they are equivalent.

Q: What are some common properties of multiplication that can help me simplify expressions?

A: Some common properties of multiplication that can help you simplify expressions include the commutative property, which states that the order of the numbers being multiplied does not change the result, and the associative property, which states that the order in which we multiply the numbers does not change the result.

Q: How do I apply the commutative and associative properties to simplify expressions?

A: To apply the commutative and associative properties, you can rearrange the numbers being multiplied to simplify the expression. For example, [ $(3)(4)(5)\$] is equivalent to \[$ (3)(5)(4)$] and also equivalent to [$(4)(3)(5)$].

Q: What are some examples of equivalent expressions?

A: Some examples of equivalent expressions include:

[ $2x + 3}$ $ and ${$3 + 2x$
[ $(3)(4)(5)}$ $ and ${$ (3)(5)(4)$
[ $(4)(3)(5)}$ $ and [$(5)(3)(4)$

Q: How do I use equivalent expressions to solve algebraic equations?

A: To use equivalent expressions to solve algebraic equations, you can simplify the equation by applying the properties of multiplication and then solving for the variable.

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 simplifying the expression correctly
Not applying the properties of multiplication correctly
Not comparing the values of the expressions correctly

Conclusion

In conclusion, equivalent expressions are mathematical expressions that have the same value, even though they may look different. By understanding the properties of multiplication and how to apply them, you can simplify expressions and solve algebraic equations. Remember to avoid common mistakes and to compare the values of the expressions correctly.

Final Tips

Practice simplifying expressions and solving algebraic equations to become more comfortable with equivalent expressions.
Use the commutative and associative properties to simplify expressions and make them easier to work with.
Compare the values of the expressions carefully to ensure that they are equivalent.

Additional Resources

For more information on equivalent expressions, check out our previous article on the topic.
For practice problems and exercises, try using online resources or algebra textbooks.
For help with specific questions or problems, try asking a teacher or tutor for assistance.