Which Property Can Be Used To Expand The Expression $-2\left(\frac{3}{4} X+7\right$\]?A. The Associative Property B. The Commutative Property C. The Distributive Property D. The Additive Identity Property

Introduction

Algebraic expressions are a fundamental concept in mathematics, and understanding how to expand them is crucial for solving equations and inequalities. In this article, we will explore the distributive property, which is a key property used to expand algebraic expressions. We will examine the given expression $-2\left(\frac{3}{4} x+7\right)$ and determine which property can be used to expand it.

The Distributive Property

The distributive property is a fundamental property in algebra that states:

a(b + c) = ab + ac

This property allows us to distribute a single term to multiple terms inside a set of parentheses. In other words, it allows us to multiply a single term by each term inside the parentheses.

Applying the Distributive Property

Let's apply the distributive property to the given expression $-2\left(\frac{3}{4} x+7\right)$. We can see that the term $-2$ is being multiplied by the terms inside the parentheses, $\frac{3}{4} x$ and $7$.

Using the distributive property, we can expand the expression as follows:

-2\left(\frac{3}{4} x+7\right) = -2\left(\frac{3}{4} x\right) -2(7)

Now, we can simplify each term:

-2\left(\frac{3}{4} x\right) = -\frac{3}{2} x

-2(7) = -14

Therefore, the expanded expression is:

-\frac{3}{2} x - 14

Conclusion

In conclusion, the distributive property is the property that can be used to expand the expression $-2\left(\frac{3}{4} x+7\right)$. This property allows us to distribute a single term to multiple terms inside a set of parentheses, making it a fundamental concept in algebra.

Key Takeaways

• The distributive property is a fundamental property in algebra that states: a(b + c) = ab + ac
• The distributive property allows us to distribute a single term to multiple terms inside a set of parentheses.
• The distributive property can be used to expand algebraic expressions.

Practice Problems

1. Expand the expression $3\left(2x+5\right)$ using the distributive property.
2. Expand the expression $-4\left(x-2\right)$ using the distributive property.
3. Expand the expression $2\left(3x-1\right)$ using the distributive property.

Answer Key

1. $6x+15$
2. $-4x+8$
3. $6x-2$

References

• Algebraic Expressions
• Distributive Property

Additional Resources

• Algebra Tutorials
• Distributive Property Practice Problems

Final Thoughts

Introduction

The distributive property is a fundamental concept in algebra that allows us to expand algebraic expressions. In our previous article, we explored the distributive property and how it can be used to expand expressions. In this article, we will answer some frequently asked questions about the distributive property.

Q: What is the distributive property?

A: The distributive property is a fundamental property in algebra that states: a(b + c) = ab + ac. This property allows us to distribute a single term to multiple terms inside a set of parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply a single term by each term inside the parentheses. For example, if you have the expression $-2\left(\frac{3}{4} x+7\right)$, you can apply the distributive property by multiplying $-2$ by each term inside the parentheses: $-2\left(\frac{3}{4} x\right) -2(7)$.

Q: What are some common mistakes to avoid when applying the distributive property?

A: Some common mistakes to avoid when applying the distributive property include:

• Forgetting to distribute the term to each term inside the parentheses
• Not simplifying the expression after applying the distributive property
• Not following the order of operations (PEMDAS)

Q: Can the distributive property be used to expand expressions with multiple sets of parentheses?

A: Yes, the distributive property can be used to expand expressions with multiple sets of parentheses. For example, if you have the expression $-2\left(\frac{3}{4} x+7\right)+3\left(x-2\right)$, you can apply the distributive property to each set of parentheses separately: $-2\left(\frac{3}{4} x\right) -2(7)+3(x) -3(2)$.

Q: How do I know which property to use when expanding an expression?

A: To determine which property to use when expanding an expression, you need to look at the expression and identify the operation that is being performed. If the expression involves multiplying a single term by multiple terms inside a set of parentheses, the distributive property is likely the correct property to use.

Q: Can the distributive property be used to simplify expressions?

A: Yes, the distributive property can be used to simplify expressions. For example, if you have the expression $-2\left(\frac{3}{4} x+7\right)$, you can apply the distributive property to simplify the expression: $-\frac{3}{2} x - 14$.

Q: What are some real-world applications of the distributive property?

A: The distributive property has many real-world applications, including:

• Simplifying algebraic expressions in physics and engineering
• Solving systems of equations in economics and finance
• Modeling population growth and decay in biology and medicine

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that allows us to expand algebraic expressions. By understanding and applying the distributive property, we can simplify complex expressions and solve equations and inequalities. We hope this article has answered some of your frequently asked questions about the distributive property.

Key Takeaways

• The distributive property is a fundamental property in algebra that states: a(b + c) = ab + ac
• The distributive property allows us to distribute a single term to multiple terms inside a set of parentheses.
• The distributive property can be used to expand expressions with multiple sets of parentheses.
• The distributive property can be used to simplify expressions.

Practice Problems

1. Expand the expression $-3\left(2x+5\right)$ using the distributive property.
2. Expand the expression $-4\left(x-2\right)$ using the distributive property.
3. Simplify the expression $-2\left(\frac{3}{4} x+7\right)$ using the distributive property.

Answer Key

1. $-6x-15$
2. $-4x+8$
3. $-\frac{3}{2} x - 14$

References

• Algebraic Expressions
• Distributive Property

Additional Resources

• Algebra Tutorials
• Distributive Property Practice Problems