The Distributive Property in Action

Oct 13, 2025 by talynt.org 1 views

The equation $3(x+7)-13=3 \cdot x+3 \cdot 7-13$ is a simple algebraic expression that can be used to demonstrate one of the fundamental properties of algebra. In this article, we will explore the properties of algebra and determine which property is demonstrated by the given equation.

What are the Properties of Algebra?

Algebra is a branch of mathematics that deals with the study of mathematical symbols and the rules for manipulating these symbols. There are three main properties of algebra: the distributive property, the commutative property, and the associative property.

Distributive Property

The distributive property is a fundamental property of algebra that states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b+c) = ab + ac$

In other words, the distributive property allows us to distribute a single term across the terms inside the parentheses.

Commutative Property

The commutative property is a property of algebra that states that for any real numbers $a$ and $b$ , the following equation holds:

$a + b = b + a$

In other words, the commutative property allows us to swap the order of the terms in an equation without changing the result.

Associative Property

The associative property is a property of algebra that states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$(a + b) + c = a + (b + c)$

In other words, the associative property allows us to regroup the terms in an equation without changing the result.

Which Property is Demonstrated by the Equation?

Now that we have a basic understanding of the properties of algebra, let's take a closer look at the equation $3(x+7)-13=3 \cdot x+3 \cdot 7-13$ . To determine which property is demonstrated by this equation, we need to examine the structure of the equation and see if it matches any of the properties we discussed earlier.

Upon closer inspection, we can see that the equation is using the distributive property to distribute the term $3$ across the terms inside the parentheses. Specifically, the equation is using the distributive property to expand the expression $3(x+7)$ into $3x + 21$ .

The Distributive Property in Action

Let's take a closer look at the distributive property in action. When we distribute the term $3$ across the terms inside the parentheses, we get:

$3(x+7) = 3x + 21$

This is a classic example of the distributive property in action. By distributing the term $3$ across the terms inside the parentheses, we are able to expand the expression and simplify it.

Conclusion

In conclusion, the equation $3(x+7)-13=3 \cdot x+3 \cdot 7-13$ demonstrates the distributive property of algebra. The distributive property allows us to distribute a single term across the terms inside the parentheses, and this property is essential for simplifying algebraic expressions.

Key Takeaways

The distributive property is a fundamental property of algebra that allows us to distribute a single term across the terms inside the parentheses.
The distributive property is essential for simplifying algebraic expressions and is used extensively in algebra.
The equation $3(x+7)-13=3 \cdot x+3 \cdot 7-13$ demonstrates the distributive property in action.

Final Thoughts

In our previous article, we explored the distributive property of algebra and demonstrated how it can be used to simplify algebraic expressions. In this article, we will answer some of the most frequently asked questions about the distributive property.

Q: What is the distributive property?

A: The distributive property is a fundamental property of algebra that allows us to distribute a single term across the terms inside the parentheses. It states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b+c) = ab + ac$

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply the single term by each of the terms inside the parentheses. For example, if we have the expression $3(x+7)$ , we can apply the distributive property by multiplying $3$ by each of the terms inside the parentheses:

$3(x+7) = 3x + 21$

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

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

Forgetting to distribute the single term across all of the terms inside the parentheses.
Not multiplying the single term by each of the terms inside the parentheses.
Not simplifying the expression after applying the distributive property.

Q: Can I use the distributive property with fractions?

A: Yes, you can use the distributive property with fractions. For example, if we have the expression $\frac{1}{2}(x+7)$ , we can apply the distributive property by multiplying $\frac{1}{2}$ by each of the terms inside the parentheses:

$\frac{1}{2}(x+7) = \frac{1}{2}x + \frac{7}{2}$

Q: Can I use the distributive property with negative numbers?

A: Yes, you can use the distributive property with negative numbers. For example, if we have the expression $-3(x+7)$ , we can apply the distributive property by multiplying $-3$ by each of the terms inside the parentheses:

$-3(x+7) = -3x - 21$

Q: How do I know when to use the distributive property?

A: You should use the distributive property whenever you have a single term multiplied by a binomial (an expression with two terms). For example, if you have the expression $3(x+7)$ , you should use the distributive property to expand the expression.

Q: Can I use the distributive property with exponents?

A: Yes, you can use the distributive property with exponents. For example, if we have the expression $2(x^2+7)$ , we can apply the distributive property by multiplying $2$ by each of the terms inside the parentheses:

$2(x^2+7) = 2x^2 + 14$

Conclusion

In conclusion, the distributive property is a fundamental property of algebra that allows us to distribute a single term across the terms inside the parentheses. By understanding the distributive property, we can simplify algebraic expressions and solve equations more efficiently. We hope that this article has answered some of the most frequently asked questions about the distributive property.

Key Takeaways

The distributive property is a fundamental property of algebra that allows us to distribute a single term across the terms inside the parentheses.
The distributive property can be used with fractions, negative numbers, and exponents.
You should use the distributive property whenever you have a single term multiplied by a binomial.
The distributive property is essential for simplifying algebraic expressions and solving equations.

Final Thoughts

In this article, we have answered some of the most frequently asked questions about the distributive property. We hope that this article has been helpful in understanding the distributive property and how to apply it in algebra.