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Introduction:

In mathematics, the distributive law is one of the fundamental laws that govern the way we perform arithmetic operations, especially multiplication and addition. It is a rule that helps us to simplify expressions involving these operations, making it easier to solve equations and manipulate mathematical formulas. The distributive law is used in a wide range of mathematical fields, from basic arithmetic to advanced algebra, and is a key concept for understanding mathematical operations.

The distributive law states that when multiplying a number by a sum or difference of numbers, we can distribute the multiplication over each term in the sum or difference and then perform the addition or subtraction. For example, if we have the expression 2(a + b), we can distribute the 2 over the terms in the parentheses and write it as 2a + 2b. Similarly, if we have the expression 3(x – 2y), we can distribute the 3 and write it as 3x – 6y. This property holds true for any two numbers and any set of numbers that can be combined with addition and subtraction.

In this article, we will explore the distributive law in more detail, including its definition, how to apply it, and some examples to help illustrate its use. We will also provide a quiz at the end of the article to test your understanding of the topic.

Definition:

The distributive law states that for any numbers a, b, and c, we have:

a x (b + c) = (a x b) + (a x c)

This law tells us how we can distribute a multiplication operation over an addition operation. We can multiply a factor a by the sum of two or more terms (b + c) by first multiplying a by each term in the sum and then adding the results together.

Examples:

Let’s look at some examples to see how the distributive law works in practice.

Example 1:

Simplify the expression 3(x + 4).

Using the distributive law, we can write:

3(x + 4) = 3x + 3(4) = 3x + 12.

Example 2:

Simplify the expression 2(3x – 5).

Using the distributive law, we can write:

2(3x – 5) = 2(3x) – 2(5) = 6x – 10.

Example 3:

Simplify the expression 5(2x + 3y).

Using the distributive law, we can write:

5(2x + 3y) = 5(2x) + 5(3y) = 10x + 15y.

Example 4:

Simplify the expression (x + y)(2x – 3y).

Using the distributive law twice, we can write:

(x + y)(2x – 3y) = x(2x – 3y) + y(2x – 3y) = 2x^2 – 3xy + 2xy – 3y^2 = 2x^2 – xy – 3y^2.

Example 5:

Simplify the expression 4(x – 3) – 3(2x + 5).

Using the distributive law, we can write:

4(x – 3) – 3(2x + 5) = 4x – 12 – 6x – 15 = -2x – 27.

More Examples!

• Simplify the expression 2(x + 3y) – 5(2x – y) + 4x. We can use the distributive law to multiply the terms outside of the parentheses by each term inside the parentheses: 2(x) + 2(3y) – 5(2x) + 5(y) + 4x = 2x + 6y – 10x + 5y + 4x = -4x + 11y
• Simplify the expression 4(2x – 5) + 2(x + 1). Using the distributive law, we can multiply the terms outside of the parentheses by each term inside: 4(2x) – 4(5) + 2(x) + 2(1) = 8x – 20 + 2x + 2 = 10x – 18
• Simplify the expression 3(2x – y) – 2(3x + 2y). Using the distributive law, we can multiply the terms outside of the parentheses by each term inside: 3(2x) – 3(y) – 2(3x) – 2(2y) = 6x – 3y – 6x – 4y = -7y
• Simplify the expression 5(x + 2) + 2(4x – 3). Using the distributive law, we can multiply the terms outside of the parentheses by each term inside: 5(x) + 5(2) + 2(4x) – 2(3) = 5x + 10 + 8x – 6 = 13x + 4
• Simplify the expression 3a(b – c) – 2b(c – a) + 4c(a – b). Using the distributive law, we can multiply the terms outside of the parentheses by each term inside: 3a(b) – 3a(c) – 2b(c) + 2b(a) + 4c(a) – 4c(b) = 3ab – 3ac – 2bc + 2ba + 4ca – 4cb = 5ba – 3ac – 2bc – 4cb + 4ca = 5ba – 7bc + 4ca – 3ac

Quiz:

1. What is the distributive law? a) a + b = b + a b) a(b + c) = ab + ac c) a(b – c) = ab – ac
2. Simplify the expression 2(x + 3y). a) 2x + 3y b) 2x – 3y c) 2x + 6y
3. Simplify the expression 3(2x – 4). a) 6x – 12 b) 6x + 12 c) 3x – 8
4. Simplify the expression (x + y)(3x – 2y). a) 3x^2 – 2xy + y^2 b) 3x^2 + 2xy + y^2 c) 3x^2 – 2xy – y^2
5. Simplify the expression 4(2x + 5) – 3(5x – 2). Using the distributive law, we can write: 4(2x + 5) – 3(5x – 2) = 8x + 20 – 15x + 6 = -7x + 26.
6. Simplify the expression 2(3a + 2b) – 3(a – 4b). a) 3a + 10b b) 3a – 2b c) 6a – 5b
7. Simplify the expression 5x(2y + 3z). a) 10xy + 15xz b) 10x + 15y + 15z c) 10xy – 15xz
8. Simplify the expression 4(a – 2b) – 3(2a + 3b). a) a – 14b b) -2a – 7b c) -2a – 14b
9. Simplify the expression (x + 2)(3x – 4) – x(2x – 1). a) 3x^2 – 7x – 8 b) 3x^2 – 7x + 8 c) 3x^2 + 7x – 8
10. Simplify the expression 2x(x – 3) + 3(x – 2). a) 2x^2 – 3x – 6 b) 2x^2 – 3x + 6 c) 2x^2 + 3x – 6

Answers to the quiz:

1. b) a(b + c) = ab + ac
2. c) 2x + 6y
3. a) 6x – 12
4. a) 3x^2 – 2xy + y^2
5. c) -7x + 26
6. c) 6a – 5b
7. a) 10xy + 15xz
8. c) -2a – 14b
9. a) 3x^2 – 7x – 8
10. b) 2x^2 – 3x + 6

Conclusion:

The distributive law is a powerful tool in mathematics that allows us to simplify complex expressions involving multiplication and addition. By using this law, we can save time and effort in solving equations and manipulating expressions. It is an essential concept in algebra and is used extensively in a variety of mathematical fields, including calculus, number theory, and geometry.

To effectively apply the distributive law, it is important to have a good understanding of the basic principles and rules involved. This article has provided an introduction to the distributive law, including its definition and examples of how to apply it. We have demonstrated how the distributive law can be used to simplify expressions, making them easier to solve and manipulate.

To solidify your understanding of the distributive law, we also provided a quiz at the end of the article. By practicing the distributive law and testing your knowledge with the quiz, you can gain confidence in your ability to apply this important concept in mathematics.

In conclusion, the distributive law is a fundamental concept in mathematics that is essential for anyone studying algebra or other advanced mathematical topics. With a solid understanding of this law, you can simplify complex expressions and make math problems more manageable. By continuing to practice and build your knowledge of the distributive law, you will be well-prepared for success in any area of mathematics.