Which Expression Is Equivalent To The Expression $\frac{2}{3}(9z+6)-7z$?A. $4-z$B. $\frac{4z}{3}+4$C. $6z+6-\frac{14z}{3}$D. $\frac{18z}{3}-\frac{12}{3}-\frac{14z}{3}$

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 process of simplifying algebraic expressions, with a focus on the given expression $\frac{2}{3}(9z+6)-7z$. We will break down the expression into manageable parts, apply the distributive property, and combine like terms to arrive at the simplified form.

Understanding the Given Expression

The given expression is $\frac{2}{3}(9z+6)-7z$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We will also need to combine like terms, which involves adding or subtracting terms with the same variable and coefficient.

Step 1: Apply the Distributive Property

The first step in simplifying the expression is to apply the distributive property to the term $\frac{2}{3}(9z+6)$. This involves multiplying the coefficient $\frac{2}{3}$ by each term inside the parentheses.

$\frac{2}{3}(9z+6) = \frac{2}{3} \cdot 9z + \frac{2}{3} \cdot 6$

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

$\frac{2}{3} \cdot 9z + \frac{2}{3} \cdot 6 = 6z + 4$

Step 2: Simplify the Expression

Now that we have applied the distributive property, we can simplify the expression by combining like terms. The expression now becomes:

$6z + 4 - 7z$

We can combine the like terms $6z$ and $-7z$ by adding their coefficients:

$6z - 7z = -z$

So, the simplified expression is:

$-z + 4$

Step 3: Rewrite the Expression in Standard Form

The simplified expression $-z + 4$ is already in standard form, but we can rewrite it in a more conventional form by combining the constant term with the variable term:

$-z + 4 = 4 - z$

Conclusion

In conclusion, the expression $\frac{2}{3}(9z+6)-7z$ is equivalent to the expression $4 - z$. We arrived at this simplified form by applying the distributive property and combining like terms. This process demonstrates the importance of simplifying algebraic expressions, as it allows us to rewrite complex expressions in a more manageable and easier-to-understand form.

Answer Key

The correct answer is:

A. $4 - z$

Additional Examples

To further illustrate the process of simplifying algebraic expressions, let's consider a few additional examples:

• $\frac{3}{4}(2x-5) + 2x$
• $\frac{2}{5}(3y+2) - 2y$
• $\frac{4}{3}(x+2) - 3x$

These examples demonstrate the application of the distributive property and the combination of like terms to simplify algebraic expressions.

Tips and Tricks

When simplifying algebraic expressions, it's essential to:

• Apply the distributive property to each term inside the parentheses.
• Combine like terms by adding or subtracting their coefficients.
• Rewrite the expression in standard form by combining the constant term with the variable term.

By following these tips and tricks, you can simplify algebraic expressions with ease and arrive at the correct solution.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes. Here are a few common mistakes to avoid:

• Failing to apply the distributive property to each term inside the parentheses.
• Not combining like terms correctly.
• Rewriting the expression in a non-standard form.

By being aware of these common mistakes, you can avoid them and arrive at the correct solution.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression $\frac{2}{3}(9z+6)-7z$. We applied the distributive property and combined like terms to arrive at the simplified form. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This means that we can multiply a single term by each term inside the parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply the coefficient by each term inside the parentheses. For example, if we have the expression $\frac{2}{3}(9z+6)$, we would multiply the coefficient $\frac{2}{3}$ by each term inside the parentheses: $\frac{2}{3} \cdot 9z + \frac{2}{3} \cdot 6$.

Q: What are like terms?

A: Like terms are terms that have the same variable and coefficient. For example, $3x$ and $5x$ are like terms because they both have the variable $x$ and the coefficient $3$ and $5$ respectively.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract their coefficients. For example, if we have the expression $3x + 5x$, we would combine the like terms by adding their coefficients: $3x + 5x = 8x$.

Q: What is the standard form of an algebraic expression?

A: The standard form of an algebraic expression is a form in which the terms are arranged in descending order of their exponents. For example, the expression $3x^2 + 2x + 1$ is in standard form.

Q: How do I rewrite an expression in standard form?

A: To rewrite an expression in standard form, simply arrange the terms in descending order of their exponents. For example, if we have the expression $2x + 3x^2$, we would rewrite it in standard form by arranging the terms in descending order of their exponents: $3x^2 + 2x$.

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

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

• Failing to apply the distributive property to each term inside the parentheses.
• Not combining like terms correctly.
• Rewriting the expression in a non-standard form.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also try simplifying expressions on your own and then checking your work with a calculator or a friend.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the distributive property and combining like terms, we can rewrite complex expressions in a more manageable and easier-to-understand form. With practice and patience, you can master the art of simplifying algebraic expressions and arrive at the correct solution.

Additional Resources

For further practice and review, you can try the following resources:

• Khan Academy: Algebra
• Mathway: Algebra
• IXL: Algebra

Final Tips

When simplifying algebraic expressions, remember to:

• Apply the distributive property to each term inside the parentheses.
• Combine like terms by adding or subtracting their coefficients.
• Rewrite the expression in standard form by arranging the terms in descending order of their exponents.

By following these tips and practicing regularly, you can become proficient in simplifying algebraic expressions and tackle even the most complex problems with confidence.