Distributive Property

• What Is Distributive Property?
• Verification of the Distributive Property
• How to Use the Distributive Property
• Solved Examples on Distributive Property
• Practice Problems on Distributive Property
• Frequently Asked Questions on Distributive Property

Introduction:

The distributive property is a fundamental concept in mathematics, specifically in algebra. It allows us to simplify expressions and solve equations by breaking down terms and distributing them across addition or subtraction operations. This property works for both multiplication and division, and it is an essential tool in solving various mathematical problems. In this article, we will explore the definition of the distributive property, provide examples to illustrate its application, discuss important facts about it, and answer frequently asked questions to enhance your understanding of this important mathematical concept.

Definition of the Distributive Property:

The distributive property states that for any three numbers a, b, and c, the expression a × (b + c) is equal to (a × b) + (a × c). In simpler terms, it means that when we multiply a number by the sum of two other numbers, it is the same as multiplying the first number individually by the other two numbers and then adding the products together. This property can also be applied to subtraction. For any three numbers a, b, and c, the expression a × (b - c) is equal to (a × b) - (a × c). The distributive property can also be used in reverse, where we can distribute a common factor outside of a parenthesis to simplify expressions.

Verification of the Distributive Property:

To verify the distributive property, we can choose any three numbers and test it using both multiplication and addition. Let's consider the numbers 2, 3, and 4.

Using the distributive property for multiplication, we have: 2 × (3 + 4) = (2 × 3) + (2 × 4) 2 × 7 = 6 + 8 14 = 14

Similarly, we can use the distributive property for subtraction: 2 × (3 - 4) = (2 × 3) - (2 × 4) 2 × (-1) = 6 - 8 -2 = -2

From these examples, we can see that the distributive property holds true. The distributive property is a fundamental rule in mathematics, and its verification is crucial to establish its validity.

How to Use the Distributive Property:

To use the distributive property, follow these steps:

1. Identify the expression that needs to be simplified or solved.
2. Look for pairs of terms that are connected by either addition or subtraction within a parenthesis.
3. Distribute the number outside the parenthesis to each term inside, applying the appropriate operation (multiplication or division).
4. Perform the necessary calculations to simplify the expression.

Let's illustrate the steps with some examples:

Example 1: Simplify the expression 3 × (2 + 5).

Step 1: The expression that needs to be simplified is 3 × (2 + 5). Step 2: There is a pair of terms connected by addition within the parenthesis. Step 3: Distribute the number 3 to each term inside the parenthesis. 3 × 2 + 3 × 5 Step 4: Perform the calculations. 6 + 15 = 21

Therefore, the simplified expression is 21.

Example 2: Simplify the expression 4 × (3 - 2).

Step 1: The expression that needs to be simplified is 4 × (3 - 2). Step 2: There is a pair of terms connected by subtraction within the parenthesis. Step 3: Distribute the number 4 to each term inside the parenthesis. 4 × 3 - 4 × 2 Step 4: Perform the calculations. 12 - 8 = 4

Therefore, the simplified expression is 4.

The distributive property is not only useful for simplifying expressions, but it is also crucial in solving equations. By distributing terms, we can rearrange equations and solve for unknown variables.

Solved Examples on Distributive Property (400 words): Let's solve some examples to further understand the application of the distributive property.

Example 1: Solve the equation 2(x + 3) = 14.

Step 1: Start by distributing the number 2 to each term inside the parenthesis. 2 × x + 2 × 3 = 14 2x + 6 = 14

Step 2: Simplify the equation. 2x = 14 - 6 2x = 8

Step 3: Isolate the variable by dividing both sides of the equation by the coefficient of x. 2x/2 = 8/2 x = 4

Therefore, the solution to the equation is x = 4.

Example 2: Solve the equation 3(2x - 4) = 15.

Step 1: Start by distributing the number 3 to each term inside the parenthesis. 3 × 2x - 3 × 4 = 15 6x - 12 = 15

Step 2: Simplify the equation. 6x = 15 + 12 6x = 27

Step 3: Isolate the variable by dividing both sides of the equation by the coefficient of x. 6x/6 = 27/6 x = 4.5

Therefore, the solution to the equation is x = 4.5.

Practice Problems on Distributive Property:

Let's practice further by solving some problems using the distributive property.

Problem 1: Simplify the expression 5 × (2 + 7).

Step 1: The expression that needs to be simplified is 5 × (2 + 7). Step 2: There is a pair of terms connected by addition within the parenthesis. Step 3: Distribute the number 5 to each term inside the parenthesis. 5 × 2 + 5 × 7 Step 4: Perform the calculations. 10 + 35 = 45

Therefore, the simplified expression is 45.

Problem 2: Simplify the expression 2 × (3 - 6).

Step 1: The expression that needs to be simplified is 2 × (3 - 6). Step 2: There is a pair of terms connected by subtraction within the parenthesis. Step 3: Distribute the number 2 to each term inside the parenthesis. 2 × 3 - 2 × 6 Step 4: Perform the calculations. 6 - 12 = -6

Therefore, the simplified expression is -6.

Problem 3: Solve the equation 3(x + 5) = 36.

Step 1: Start by distributing the number 3 to each term inside the parenthesis. 3x + 3 × 5 = 36 3x + 15 = 36

Step 2: Simplify the equation. 3x = 36 - 15 3x = 21

Step 3: Isolate the variable by dividing both sides of the equation by the coefficient of x. 3x/3 = 21/3 x = 7

Therefore, the solution to the equation is x = 7.

Problem 4: Solve the equation 4(2x - 3) = 8.

Step 1: Start by distributing the number 4 to each term inside the parenthesis. 4 × 2x - 4 × 3 = 8 8x - 12 = 8

Step 2: Simplify the equation. 8x = 8 + 12 8x = 20

Step 3: Isolate the variable by dividing both sides of the equation by the coefficient of x. 8x/8 = 20/8 x = 2.5

Therefore, the solution to the equation is x = 2.5.

Frequently Asked Questions on Distributive Property:

To further enhance your understanding of the distributive property, let's answer some frequently asked questions.

Q1. Is the distributive property only applicable for multiplication and addition?Q1. Is the distributive property only applicable for multiplication and addition?

A1. No, the distributive property applies to both multiplication and division, as well as addition and subtraction. When distributing a number outside of a parenthesis, we can multiply or divide the terms inside accordingly.

Q2. Can the distributive property be used in reverse?

A2. Yes, the distributive property can be used in reverse. Instead of distributing a number outside of a parenthesis, we can factor out a common factor from an expression and simplify it.

Q3. Can the distributive property be applied to variables?

A3. Yes, the distributive property can be applied to variables. The property works for any three numbers, including variables. The distributive property allows us to simplify expressions and solve equations involving variables.

Q4. Can the distributive property be used with more than two terms inside the parenthesis?Q4. Can the distributive property be used with more than two terms inside the parenthesis?

A4. Yes, the distributive property can be used with more than two terms inside the parenthesis. In such cases, we distribute the number outside the parenthesis to each term individually.

Q5. Can the distributive property be used with fractions and decimals?

A5. Yes, the distributive property can be used with fractions and decimals. The property applies to all types of numbers, including fractions and decimals. We can distribute the number outside the parenthesis to each term, whether they are fractions, decimals, or whole numbers.

Q6. Is the distributive property a specific rule, or is it based on other mathematical principles?Q6. Is the distributive property a specific rule, or is it based on other mathematical principles?

A6. The distributive property is a specific mathematical rule that is based on the principles of arithmetic and algebra. It is a fundamental property that allows us to simplify expressions.