Use The Distributive Property To Combine Like Terms And Simplify The Expression: $10x + 3(x - 15$\]A. $13x - 15$B. $13x + 45$C. $-32x$D. $13x - 45$

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside. In this article, we will explore how to use the distributive property to combine like terms and simplify the expression $10x + 3(x - 15)$.

What is the Distributive Property?

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

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

This means that we can distribute the term $a$ to each term inside the parentheses, $b$ and $c$, and then combine the results.

Applying the Distributive Property to the Expression

Now, let's apply the distributive property to the expression $10x + 3(x - 15)$. We can start by distributing the term $3$ to each term inside the parentheses, $x$ and $-15$.

$10x + 3(x - 15) = 10x + 3x - 45$

Combining Like Terms

Now that we have expanded the expression, we can combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms, $10x$ and $3x$, which can be combined by adding their coefficients.

$10x + 3x - 45 = 13x - 45$

Simplifying the Expression

The expression $13x - 45$ is the simplified form of the original expression $10x + 3(x - 15)$. We can see that the distributive property has helped us to combine like terms and simplify the expression.

Conclusion

In this article, we have learned how to use the distributive property to combine like terms and simplify the expression $10x + 3(x - 15)$. By applying the distributive property and combining like terms, we have arrived at the simplified form of the expression, $13x - 45$. This is an important concept in algebra that can be applied to a wide range of problems.

Common Mistakes to Avoid

When applying the distributive property, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not distributing the term correctly: Make sure to distribute the term to each term inside the parentheses.
• Not combining like terms: Make sure to combine like terms by adding their coefficients.
• Not simplifying the expression: Make sure to simplify the expression by combining like terms and eliminating any unnecessary terms.

Practice Problems

Here are some practice problems to help you apply the distributive property and combine like terms:

1. Simplify the expression $2x + 4(x + 3)$.
2. Simplify the expression $3x - 2(x - 4)$.
3. Simplify the expression $5x + 2(x + 1)$.

Answer Key

1. $6x + 12$
2. $x + 6$
3. $7x + 2$

Final Thoughts

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside. 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 each term inside the parentheses with the term outside. For example, if we have the expression $2(x + 3)$, we can apply the distributive property by multiplying $2$ with each term inside the parentheses:

$2(x + 3) = 2x + 6$

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $4x$ are like terms because they both have the variable $x$ raised to the power of 1. Like terms can be combined by adding their coefficients.

Q: How do I combine like terms?

A: To combine like terms, simply add their coefficients. For example, if we have the expression $2x + 4x$, we can combine the like terms by adding their coefficients:

$2x + 4x = 6x$

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different concepts in algebra. The distributive property allows us to expand and simplify expressions by multiplying each term inside the parentheses with the term outside. The commutative property, on the other hand, states that the order of the terms in an expression does not change the value of the expression. For example, $2x + 3x = 3x + 2x$.

Q: Can I use the distributive property to simplify expressions with fractions?

A: Yes, you can use the distributive property to simplify expressions with fractions. For example, if we have the expression $\frac{1}{2}(x + 3)$, we can apply the distributive property by multiplying $\frac{1}{2}$ with each term inside the parentheses:

$\frac{1}{2}(x + 3) = \frac{1}{2}x + \frac{3}{2}$

Q: How do I apply the distributive property to expressions with exponents?

A: To apply the distributive property to expressions with exponents, simply multiply each term inside the parentheses with the term outside. For example, if we have the expression $2(x^2 + 3)$, we can apply the distributive property by multiplying $2$ with each term inside the parentheses:

$2(x^2 + 3) = 2x^2 + 6$

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

A: Yes, you can use the distributive property to simplify expressions with negative numbers. For example, if we have the expression $-2(x + 3)$, we can apply the distributive property by multiplying $-2$ with each term inside the parentheses:

$-2(x + 3) = -2x - 6$

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

A: You should use the distributive property whenever you have an expression with parentheses and you want to simplify it. The distributive property is a powerful tool that can help you simplify complex expressions and make them easier to work with.

Q: Can I use the distributive property to simplify expressions with variables on both sides?

A: Yes, you can use the distributive property to simplify expressions with variables on both sides. For example, if we have the expression $x(2x + 3)$, we can apply the distributive property by multiplying $x$ with each term inside the parentheses:

$x(2x + 3) = 2x^2 + 3x$

Q: How do I check my work when using the distributive property?

A: To check your work when using the distributive property, simply multiply each term inside the parentheses with the term outside and then combine like terms. If your answer is correct, you should be able to simplify the expression and arrive at the same answer.