Fill In The BlankConsider The Expression: $3(5x + 2y$\]. Enter An Expression (with No Spaces And In Alphabetical Order) That Shows The Sum Of Exactly Two Terms Equivalent To $3(5x + 2y$\].Type Your AnswerMultiple AnswerSelect All The...

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: $3(5x + 2y)$. We will break down the expression into its individual components, and then combine them to form a new expression that is equivalent to the original.

Understanding the Expression

The given expression is $3(5x + 2y)$. This expression consists of two main components: the coefficient $3$ and the binomial $5x + 2y$. The coefficient $3$ is a numerical value that is multiplied by the binomial $5x + 2y$.

Distributive Property

To simplify the expression, we will use the distributive property, which states that for any numbers $a$, $b$, and $c$, the following equation holds:

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

In our case, we can apply the distributive property to the expression $3(5x + 2y)$ as follows:

$3(5x + 2y) = 3 \cdot 5x + 3 \cdot 2y$

Simplifying the Expression

Now that we have applied the distributive property, we can simplify the expression further by multiplying the coefficient $3$ by each term in the binomial.

$3 \cdot 5x = 15x$

$3 \cdot 2y = 6y$

Therefore, the simplified expression is:

$15x + 6y$

Conclusion

In this article, we have simplified the algebraic expression $3(5x + 2y)$ using the distributive property. We have broken down the expression into its individual components, and then combined them to form a new expression that is equivalent to the original. The simplified expression is $15x + 6y$, which is the sum of exactly two terms.

Key Takeaways

• The distributive property is a fundamental concept in algebra that allows us to simplify complex expressions.
• To simplify an expression using the distributive property, we need to multiply the coefficient by each term in the binomial.
• The simplified expression is the sum of exactly two terms.

Practice Problems

1. Simplify the expression $2(3x + 4y)$ using the distributive property.
2. Simplify the expression $4(2x + 3y)$ using the distributive property.
3. Simplify the expression $5(2x + 3y)$ using the distributive property.

Answer Key

1. $6x + 8y$
2. $8x + 12y$
3. $10x + 15y**

Additional Resources

For more practice problems and additional resources, please visit the following websites:

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

Conclusion

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to simplify complex expressions. It states that for any numbers $a$, $b$, and $c$, the following equation holds:

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

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

A: To apply the distributive property, you need to multiply the coefficient by each term in the binomial. For example, if you have the expression $3(5x + 2y)$, you would multiply the coefficient $3$ by each term in the binomial:

$3 \cdot 5x = 15x$

$3 \cdot 2y = 6y$

Therefore, the simplified expression is $15x + 6y$.

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 simplify complex expressions by multiplying the coefficient by each term in the binomial. 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, if you have the expression $2x + 3y$, you can rearrange the terms to get $3y + 2x$, but the value of the expression remains the same.

Q: Can I simplify an expression with multiple coefficients?

A: Yes, you can simplify an expression with multiple coefficients by applying the distributive property to each coefficient separately. For example, if you have the expression $2(3x + 4y) + 5(2x + 3y)$, you would apply the distributive property to each coefficient separately:

$2 \cdot 3x = 6x$

$2 \cdot 4y = 8y$

$5 \cdot 2x = 10x$

$5 \cdot 3y = 15y$

Therefore, the simplified expression is $16x + 23y$.

Q: How do I simplify an expression with a negative coefficient?

A: To simplify an expression with a negative coefficient, you need to apply the distributive property as usual, but then multiply each term by the negative coefficient. For example, if you have the expression $-3(5x + 2y)$, you would multiply the coefficient $-3$ by each term in the binomial:

$-3 \cdot 5x = -15x$

$-3 \cdot 2y = -6y$

Therefore, the simplified expression is $-15x - 6y$.

Q: Can I simplify an expression with variables in the coefficient?

A: Yes, you can simplify an expression with variables in the coefficient by applying the distributive property as usual. For example, if you have the expression $2x(3x + 4y)$, you would multiply the coefficient $2x$ by each term in the binomial:

$2x \cdot 3x = 6x^2$

$2x \cdot 4y = 8xy$

Therefore, the simplified expression is $6x^2 + 8xy$.

Q: How do I simplify an expression with a fraction as a coefficient?

A: To simplify an expression with a fraction as a coefficient, you need to apply the distributive property as usual, but then multiply each term by the fraction. For example, if you have the expression $\frac{1}{2}(5x + 2y)$, you would multiply the coefficient $\frac{1}{2}$ by each term in the binomial:

$\frac{1}{2} \cdot 5x = \frac{5}{2}x$

$\frac{1}{2} \cdot 2y = y$

Therefore, the simplified expression is $\frac{5}{2}x + y$.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the distributive property, we can break down complex expressions into their individual components and combine them to form a new expression that is equivalent to the original. We hope that this article has provided you with a better understanding of the distributive property and how to simplify algebraic expressions.