Which Expression Is Equivalent To $(-3y-x)-(5y-8x$\]?A. $-8y-8x$B. $-8y+7x$C. $2y-7x$D. $8y+8x$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the distributive property and combining like terms. We will also apply this knowledge to solve a specific problem, which is to find the equivalent expression to $(-3y-x)-(5y-8x)$.

Understanding the 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 factor outside the parentheses. This property can be written as:

$a(b+c) = ab + ac$

where $a$, $b$, and $c$ are algebraic expressions.

Applying the Distributive Property

To simplify the expression $(-3y-x)-(5y-8x)$, we need to apply the distributive property to each term inside the parentheses. This means that we will multiply each term by the factor outside the parentheses.

$(-3y-x)-(5y-8x) = (-3y-x)(-1) + (5y-8x)(-1)$

Using the distributive property, we can expand each term as follows:

$(-3y-x)(-1) = 3y + x$

$(5y-8x)(-1) = -5y + 8x$

Now, we can combine the two expanded terms to get the simplified expression:

$3y + x - 5y + 8x$

Combining Like Terms

To simplify the expression further, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $y$ and two terms with the variable $x$.

$3y + x - 5y + 8x$

We can combine the like terms as follows:

$3y - 5y = -2y$

$x + 8x = 9x$

Now, we can combine the two simplified terms to get the final expression:

$-2y + 9x$

Comparing the Simplified Expression to the Answer Choices

Now that we have simplified the expression, we can compare it to the answer choices to see which one is equivalent.

A. $-8y-8x$

B. $-8y+7x$

C. $2y-7x$

D. $8y+8x$

Based on our simplified expression, we can see that the correct answer is:

B. $-8y+7x$

This is because our simplified expression is $-2y + 9x$, which is equivalent to $-8y + 7x$.

Conclusion

In this article, we explored the process of simplifying algebraic expressions using the distributive property and combining like terms. We applied this knowledge to solve a specific problem, which was to find the equivalent expression to $(-3y-x)-(5y-8x)$. We compared our simplified expression to the answer choices and found that the correct answer is B. $-8y+7x$. This article provides a step-by-step guide on how to simplify algebraic expressions and apply the distributive property and combining like terms to solve problems.

Additional Tips and Resources

• To simplify algebraic expressions, start by applying the distributive property to each term inside the parentheses.
• Combine like terms by adding or subtracting the coefficients of the same variable.
• Use the distributive property to expand expressions and simplify them further.
• Practice simplifying algebraic expressions by working through examples and exercises.

References

• Algebraic Expressions
• Distributive Property
• Combining Like Terms

Frequently Asked Questions

• 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 factor outside the parentheses.
• Q: How do I combine like terms? A: To combine like terms, add or subtract the coefficients of the same variable.
• Q: What is the difference between the distributive property and combining like terms? A: The distributive property is used to expand expressions, while combining like terms is used to simplify expressions by adding or subtracting the coefficients of the same variable.
  Frequently Asked Questions: Simplifying Algebraic Expressions =============================================================

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 factor outside the parentheses. This property can be written as:

$a(b+c) = ab + ac$

where $a$, $b$, and $c$ are algebraic expressions.

Q: How do I apply the distributive property?

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

$2(x+3) = 2x + 6$

Q: What is the difference between the distributive property and combining like terms?

A: The distributive property is used to expand expressions, while combining like terms is used to simplify expressions by adding or subtracting the coefficients of the same variable. For example:

$2x + 3x = 5x$

In this example, we combined the like terms $2x$ and $3x$ to get $5x$.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the same variable. For example:

$2x + 3x = 5x$

$2y - 3y = -y$

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying expressions. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify expressions with variables and constants?

A: To simplify expressions with variables and constants, follow these steps:

1. Apply the distributive property to expand any expressions inside parentheses.
2. Combine like terms by adding or subtracting the coefficients of the same variable.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

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

A: Some common mistakes to avoid when simplifying expressions include:

• Forgetting to apply the distributive property to expressions inside parentheses.
• Not combining like terms correctly.
• Not following the order of operations.
• Not simplifying exponential expressions correctly.

Q: How can I practice simplifying expressions?

A: There are many ways to practice simplifying expressions, including:

• Working through examples and exercises in a textbook or online resource.
• Using online tools or calculators to simplify expressions.
• Creating your own expressions to simplify and solving them.
• Joining a study group or working with a tutor to practice simplifying expressions.

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

A: Simplifying expressions has many real-world applications, including:

• Science: Simplifying expressions is used to solve equations and model real-world phenomena.
• Engineering: Simplifying expressions is used to design and optimize systems.
• Finance: Simplifying expressions is used to calculate interest rates and investment returns.
• Computer Science: Simplifying expressions is used to optimize algorithms and solve problems.

Conclusion

Simplifying expressions is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can simplify expressions with confidence and accuracy. Remember to apply the distributive property, combine like terms, and follow the order of operations to simplify expressions. With practice and patience, you can become proficient in simplifying expressions and tackle even the most complex problems.