Which Expressions Are Equivalent To $3\left(\frac{2}{3} P+3-\frac{1}{3} P-5\right$\]? Choose ALL That Apply:A. $3\left(\frac{1}{3} P-2\right$\]B. $3\left(\frac{1}{3} P+3-5\right$\]C. $p-6$D. $p-2$E.

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given problem: $3\left(\frac{2}{3} p+3-\frac{1}{3} p-5\right)$. We will examine the different options provided and determine which ones are equivalent to the given expression.

Understanding the Given Expression

The given expression is $3\left(\frac{2}{3} p+3-\frac{1}{3} p-5\right)$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Multiply the result by 3.

Step 1: Evaluate the Expressions Inside the Parentheses

Let's start by evaluating the expressions inside the parentheses:

$\frac{2}{3} p+3-\frac{1}{3} p-5$

We can combine like terms by adding or subtracting the coefficients of the same variable:

$\frac{2}{3} p-\frac{1}{3} p+3-5$

Simplifying further, we get:

$\frac{1}{3} p-2$

Step 2: Multiply the Result by 3

Now that we have simplified the expression inside the parentheses, we can multiply the result by 3:

$3\left(\frac{1}{3} p-2\right)$

Analyzing the Options

Let's examine the options provided and determine which ones are equivalent to the simplified expression:

Option A: $3\left(\frac{1}{3} p-2\right)$

This option is equivalent to the simplified expression we obtained in Step 2.

Option B: $3\left(\frac{1}{3} p+3-5\right)$

To simplify this expression, we need to follow the order of operations:

$3\left(\frac{1}{3} p+3-5\right)$

$= 3\left(\frac{1}{3} p-2\right)$

This option is also equivalent to the simplified expression we obtained in Step 2.

Option C: $p-6$

To determine if this option is equivalent, we need to simplify the expression:

$3\left(\frac{1}{3} p-2\right)$

$= p-6$

This option is equivalent to the simplified expression we obtained in Step 2.

Option D: $p-2$

This option is not equivalent to the simplified expression we obtained in Step 2.

Option E: Not Provided

Since Option E is not provided, we cannot determine if it is equivalent to the simplified expression.

Conclusion

In conclusion, the options that are equivalent to the given expression $3\left(\frac{2}{3} p+3-\frac{1}{3} p-5\right)$ are:

• Option A: $3\left(\frac{1}{3} p-2\right)$
• Option B: $3\left(\frac{1}{3} p+3-5\right)$
• Option C: $p-6$

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given problem: $3\left(\frac{2}{3} p+3-\frac{1}{3} p-5\right)$. We examined the different options provided and determined which ones are equivalent to the given expression. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in simplifying algebraic expressions.

Q&A Guide

Q: What is the order of operations in algebra?

A: The order of operations in algebra is PEMDAS, which 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?

A: To simplify an algebraic expression, follow these steps:

1. Evaluate any expressions inside parentheses.
2. Combine like terms by adding or subtracting the coefficients of the same variable.
3. Simplify any exponential expressions.
4. Multiply or divide any remaining terms.

Q: What is a like term?

A: A like term is a term that has the same variable raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the same variable. For example, $2x + 5x = 7x$ because the coefficients of $x$ are added together.

Q: What is the distributive property?

A: The distributive property is a rule that allows you to multiply a single term by multiple terms inside parentheses. For example, $3(x + 2) = 3x + 6$.

Q: How do I simplify an expression with parentheses?

A: To simplify an expression with parentheses, follow these steps:

1. Evaluate any expressions inside the innermost parentheses first.
2. Work your way outwards, evaluating any expressions inside the next set of parentheses.
3. Finally, simplify any remaining terms.

Q: What is the difference between an expression and an equation?

A: An expression is a group of terms that are combined using mathematical operations. An equation is a statement that says two expressions are equal. For example, $2x + 3 = 5$ is an equation because it says that the expression $2x + 3$ is equal to the expression $5$.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill to master in mathematics. By following the order of operations and combining like terms, you can simplify even the most complex expressions. We hope this Q&A guide has helped you better understand the concepts and techniques involved in simplifying algebraic expressions.

Common Algebraic Expressions and Their Simplifications

Here are some common algebraic expressions and their simplifications:

• $2x + 3x = 5x$
• $3(x + 2) = 3x + 6$
• $2(x - 3) = 2x - 6$
• $x + 2x = 3x$
• $x - 2x = -x$

Practice Problems

Here are some practice problems to help you reinforce your understanding of simplifying algebraic expressions:

1. Simplify the expression: $3(2x + 5) - 2(x - 3)$
2. Simplify the expression: $2(x + 2) + 3(x - 1)$
3. Simplify the expression: $x + 2x + 3x$
4. Simplify the expression: $x - 2x - 3x$
5. Simplify the expression: $3(x + 2) - 2(x - 1)$

We hope these practice problems help you better understand the concepts and techniques involved in simplifying algebraic expressions.