Which Expression Is Equivalent To The Expression $5b + 16$?A. $3b - 6 + 2b + 22$B. $5b - 6 + B + 22$C. $2b + 4 + 2b + 12$D. $7b + 3 - 2b + 9$

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 identifying equivalent expressions. We will examine a specific problem, where we need to determine which expression is equivalent to the expression $5b + 16$. This problem requires us to apply our knowledge of algebraic properties, such as the distributive property and the commutative property.

Understanding Algebraic Properties

Before we dive into the problem, let's review some essential algebraic properties that will help us simplify expressions.

• Distributive Property: The distributive property states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.
• Commutative Property: The commutative property states that for any numbers $a$ and $b$, $a + b = b + a$ and $ab = ba$.
• Associative Property: The associative property states that for any numbers $a$, $b$, and $c$, $(a + b) + c = a + (b + c)$.

The Problem

We are given the expression $5b + 16$ and need to determine which of the following expressions is equivalent to it:

A. $3b - 6 + 2b + 22$ B. $5b - 6 + b + 22$ C. $2b + 4 + 2b + 12$ D. $7b + 3 - 2b + 9$

Step 1: Simplify Expression A

Let's start by simplifying expression A: $3b - 6 + 2b + 22$.

Using the distributive property, we can rewrite the expression as:

$3b + 2b - 6 + 22$

Combining like terms, we get:

$5b + 16$

This expression is equivalent to the original expression $5b + 16$.

Step 2: Simplify Expression B

Next, let's simplify expression B: $5b - 6 + b + 22$.

Using the commutative property, we can rewrite the expression as:

$5b + b - 6 + 22$

Combining like terms, we get:

$6b + 16$

This expression is not equivalent to the original expression $5b + 16$.

Step 3: Simplify Expression C

Now, let's simplify expression C: $2b + 4 + 2b + 12$.

Using the distributive property, we can rewrite the expression as:

$2b + 2b + 4 + 12$

Combining like terms, we get:

$4b + 16$

This expression is not equivalent to the original expression $5b + 16$.

Step 4: Simplify Expression D

Finally, let's simplify expression D: $7b + 3 - 2b + 9$.

Using the commutative property, we can rewrite the expression as:

$7b - 2b + 3 + 9$

Combining like terms, we get:

$5b + 12$

This expression is not equivalent to the original expression $5b + 16$.

Conclusion

In conclusion, the expression that is equivalent to the expression $5b + 16$ is:

A. $3b - 6 + 2b + 22$

This expression simplifies to $5b + 16$, which is equivalent to the original expression.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Use the distributive property: The distributive property is a powerful tool for simplifying expressions. Use it to rewrite expressions and combine like terms.
• Use the commutative property: The commutative property is another essential property for simplifying expressions. Use it to rewrite expressions and combine like terms.
• Combine like terms: Combining like terms is a crucial step in simplifying expressions. Make sure to combine all like terms to get the simplest expression possible.
• Check your work: Always check your work to ensure that the expression you simplified is equivalent to the original expression.

By following these tips and tricks, you can become a master of simplifying algebraic expressions and solving problems like the one presented in this article.