Which Expressions Are Equivalent To $-\frac{1}{2}(6-4+10x$\]?Select All That Apply.A. $5x + 1$ B. $5x - 1$ C. $4x$ D. $-3 + 2 - 5x$ E. $3 + 2 - 5x$ F. $-5x - 1$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the concept of equivalent expressions and learn how to simplify complex algebraic expressions. We will focus on the given expression $-\frac{1}{2}(6-4+10x)$ and determine which of the provided options are equivalent to it.

Understanding Equivalent Expressions

Equivalent expressions are algebraic expressions that have the same value, but may be written in different forms. For example, the expressions $2x + 3$ and $x + 2 + x + 1$ are equivalent because they both represent the same value. In this article, we will learn how to identify equivalent expressions and simplify complex algebraic expressions.

Simplifying the Given Expression

The given expression is $-\frac{1}{2}(6-4+10x)$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $6-4+10x = 2+10x$
2. Multiply the result by $-\frac{1}{2}$: $-\frac{1}{2}(2+10x) = -1-5x$

Analyzing the Options

Now that we have simplified the given expression, let's analyze the options:

A. $5x + 1$ B. $5x - 1$ C. $4x$ D. $-3 + 2 - 5x$ E. $3 + 2 - 5x$ F. $-5x - 1$

Option A: $5x + 1$

This option is not equivalent to the simplified expression $-1-5x$. To see why, let's substitute $x=0$ into both expressions:

• $-1-5x = -1-5(0) = -1$
• $5x + 1 = 5(0) + 1 = 1$

As we can see, the two expressions have different values when $x=0$, so option A is not equivalent to the simplified expression.

Option B: $5x - 1$

This option is also not equivalent to the simplified expression $-1-5x$. To see why, let's substitute $x=0$ into both expressions:

• $-1-5x = -1-5(0) = -1$
• $5x - 1 = 5(0) - 1 = -1$

Although the two expressions have the same value when $x=0$, they are not equivalent because they have different coefficients for the variable $x$. Therefore, option B is not equivalent to the simplified expression.

Option C: $4x$

This option is not equivalent to the simplified expression $-1-5x$. To see why, let's substitute $x=0$ into both expressions:

• $-1-5x = -1-5(0) = -1$
• $4x = 4(0) = 0$

As we can see, the two expressions have different values when $x=0$, so option C is not equivalent to the simplified expression.

Option D: $-3 + 2 - 5x$

This option is equivalent to the simplified expression $-1-5x$. To see why, let's simplify the expression:

$-3 + 2 - 5x = -1 - 5x$

As we can see, the two expressions are equivalent.

Option E: $3 + 2 - 5x$

This option is not equivalent to the simplified expression $-1-5x$. To see why, let's simplify the expression:

$3 + 2 - 5x = 5 - 5x$

As we can see, the two expressions have different coefficients for the variable $x$, so option E is not equivalent to the simplified expression.

Option F: $-5x - 1$

This option is equivalent to the simplified expression $-1-5x$. To see why, let's simplify the expression:

$-5x - 1 = -1 - 5x$

As we can see, the two expressions are equivalent.

Conclusion

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Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying algebraic expressions. The acronym PEMDAS stands for:

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 an algebraic expression with multiple operations?

A: To simplify an algebraic expression with multiple operations, follow the order of operations (PEMDAS):

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponential expressions.
3. Evaluate any multiplication and division operations from left to right.
4. Finally, evaluate any addition and subtraction operations from left to right.

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

A: Equivalent expressions are algebraic expressions that have the same value, but may be written in different forms. Equivalent values, on the other hand, are values that have the same numerical value, but may be expressed in different ways.

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

A: To determine if two expressions are equivalent, follow these steps:

1. Simplify both expressions using the order of operations (PEMDAS).
2. Compare the simplified expressions to see if they have the same value.
3. If the simplified expressions have the same value, then the original expressions are equivalent.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it helps us:

1. Solve equations and inequalities more easily.
2. Understand the relationships between variables and constants.
3. Make calculations more efficient and accurate.
4. Identify patterns and relationships in algebraic expressions.

Q: Can you provide examples of equivalent expressions?

A: Yes, here are some examples of equivalent expressions:

• $2x + 3$ and $x + 2 + x + 1$
• $5x - 2$ and $x + 5x - 2$
• $x^2 + 4x + 4$ and $(x + 2)^2$

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, follow these steps:

1. Rewrite the expression with a positive exponent by taking the reciprocal of the base.
2. Simplify the expression using the order of operations (PEMDAS).

For example, to simplify the expression $x^{-2}$, we can rewrite it as $\frac{1}{x^2}$.

Q: Can you provide examples of expressions with negative exponents?

A: Yes, here are some examples of expressions with negative exponents:

• $x^{-2} = \frac{1}{x^2}$
• $y^{-3} = \frac{1}{y^3}$
• $z^{-4} = \frac{1}{z^4}$

Conclusion

In this article, we answered some frequently asked questions about simplifying algebraic expressions. We covered topics such as the order of operations (PEMDAS), equivalent expressions, and negative exponents. We hope this article has been helpful in clarifying any doubts you may have had about simplifying algebraic expressions.