Which Of The Following Expressions Are Equivalent To $40x + 30y$? Choose ALL That Apply.A. $5x(8 + 6$\]B. $5(8x + 6y$\]C. $10(40x + 30y$\]D. $10(4x + 3y$\]E. $2y(20x + 15$\]F. $2(20x + 15y$\]

Introduction

In algebra, equivalent expressions are those that have the same value for all possible values of the variables involved. In this article, we will explore which of the given expressions are equivalent to the expression $40x + 30y$. We will analyze each option carefully and provide a detailed explanation of why some expressions are equivalent while others are not.

Option A: $5x(8 + 6y)$

Let's start by analyzing option A: $5x(8 + 6y)$. To determine if this expression is equivalent to $40x + 30y$, we need to expand the expression using the distributive property.

$$5x(8 + 6y) = 5x \cdot 8 + 5x \cdot 6y = 40x + 30xy$$

As we can see, the expression $5x(8 + 6y)$ is not equivalent to $40x + 30y$ because it contains an additional term $30xy$ that is not present in the original expression.

Option B: $5(8x + 6y)$

Next, let's analyze option B: $5(8x + 6y)$. To determine if this expression is equivalent to $40x + 30y$, we need to expand the expression using the distributive property.

$$5(8x + 6y) = 5 \cdot 8x + 5 \cdot 6y = 40x + 30y$$

As we can see, the expression $5(8x + 6y)$ is equivalent to $40x + 30y$.

Option C: $10(40x + 30y)$

Now, let's analyze option C: $10(40x + 30y)$. To determine if this expression is equivalent to $40x + 30y$, we need to expand the expression using the distributive property.

$$10(40x + 30y) = 10 \cdot 40x + 10 \cdot 30y = 400x + 300y$$

As we can see, the expression $10(40x + 30y)$ is not equivalent to $40x + 30y$ because it contains additional terms $400x$ and $300y$ that are not present in the original expression.

Option D: $10(4x + 3y)$

Next, let's analyze option D: $10(4x + 3y)$. To determine if this expression is equivalent to $40x + 30y$, we need to expand the expression using the distributive property.

$$10(4x + 3y) = 10 \cdot 4x + 10 \cdot 3y = 40x + 30y$$

As we can see, the expression $10(4x + 3y)$ is equivalent to $40x + 30y$.

Option E: $2y(20x + 15)$

Now, let's analyze option E: $2y(20x + 15)$. To determine if this expression is equivalent to $40x + 30y$, we need to expand the expression using the distributive property.

$$2y(20x + 15) = 2y \cdot 20x + 2y \cdot 15 = 40xy + 30y$$

As we can see, the expression $2y(20x + 15)$ is not equivalent to $40x + 30y$ because it contains an additional term $40xy$ that is not present in the original expression.

Option F: $2(20x + 15y)$

Finally, let's analyze option F: $2(20x + 15y)$. To determine if this expression is equivalent to $40x + 30y$, we need to expand the expression using the distributive property.

$$2(20x + 15y) = 2 \cdot 20x + 2 \cdot 15y = 40x + 30y$$

As we can see, the expression $2(20x + 15y)$ is equivalent to $40x + 30y$.

Conclusion

In conclusion, the expressions that are equivalent to $40x + 30y$ are:

• Option B: $5(8x + 6y)$
• Option D: $10(4x + 3y)$
• Option F: $2(20x + 15y)$

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Introduction

In our previous article, we explored which of the given expressions are equivalent to the expression $40x + 30y$. We analyzed each option carefully and provided a detailed explanation of why some expressions are equivalent while others are not. In this article, we will provide a Q&A guide to help you better understand equivalent expressions in algebra.

Q: What is an equivalent expression in algebra?

A: An equivalent expression in algebra is an expression that has the same value for all possible values of the variables involved. In other words, two expressions are equivalent if they can be transformed into each other using the rules of algebra.

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

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

1. Expand both expressions using the distributive property.
2. Simplify both expressions by combining like terms.
3. Compare the simplified expressions to see if they are the same.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Here are some common mistakes to avoid when working with equivalent expressions:

• Not expanding expressions fully before simplifying them.
• Not combining like terms correctly.
• Not checking if expressions are equivalent before simplifying them.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, you need to follow these steps:

1. Identify the terms in the expression that can be multiplied together.
2. Multiply each term by the other term using the distributive property.
3. Simplify the resulting expression by combining like terms.

Q: What is the distributive property in algebra?

A: The distributive property in algebra is a rule that states that a single term can be multiplied by multiple terms. In other words, the distributive property allows you to multiply a single term by multiple terms and then combine the results.

Q: How do I use the distributive property to simplify an expression?

A: To use the distributive property to simplify an expression, you need to follow these steps:

1. Identify the terms in the expression that can be multiplied together.
2. Multiply each term by the other term using the distributive property.
3. Simplify the resulting expression by combining like terms.

Q: What are some examples of equivalent expressions?

A: Here are some examples of equivalent expressions:

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

Q: How do I determine if an expression is equivalent to a given expression?

A: To determine if an expression is equivalent to a given expression, you need to follow these steps:

1. Expand both expressions using the distributive property.
2. Simplify both expressions by combining like terms.
3. Compare the simplified expressions to see if they are the same.

Conclusion

In conclusion, equivalent expressions in algebra are expressions that have the same value for all possible values of the variables involved. By following the steps outlined in this article, you can determine if two expressions are equivalent and simplify expressions using the distributive property. Remember to avoid common mistakes and to check if expressions are equivalent before simplifying them.