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# Distributive Law|Definition & Meaning

## Definition

According to the **Distributive Law** in mathematics, carrying out one multiplication at a time is equivalent to multiplying a value by a collection of previously added numbers. For instance: **[4× (3 + 4)] = (4 × 2) + (4 × 4)**.

**What Is the Distributive Law **

A single value, along with **two** or **more values** contained within a set of **parentheses**, can be multiplied with one another using an **algebraic property** known as the **Distributive Law**.

According to the Distributive Law, to correctly calculate the result of multiplying a factor by the **sum** or **addition** of two terms, it is necessary first to perform **multiplication** on each of the **two numbers** by the factor and then to carry out the **addition operation**.

This is because the factor is multiplied by the sum or addition of both numbers. The following is a **symbolic representation** of this property:

**X (Y + Z) = XY + XZ**

Where **X**, **Y**, and **Z** each correspond to a different value.

**Distributive Property With Variables**

Take a look at the following example: **7 (3 + 5x)**

Because the two numbers contained within the **parentheses** can never be added together because they do not correspond to the same thing, the problem **cannot be simplified** anymore. We need an alternative way, and **Distributive Property** can be employed in this situation as an option.

When the Distributive Property is considered:

**(7 × 3) + (7 × 5x)**

The parenthesis has been **eliminated**, and each phrase has been increased by a **factor of 6**.

You should now be able to simplify the multiplication by only considering individual terms.

**21 + 35x**

**Expressions** that involve multiplying a number by a sum or difference can be made more straightforward by utilizing the **distributive property of multiplication**. This property states that the product of the sum or difference of a number is **equivalent** to the sum or **difference** of the multiplication of the number. In algebra, the distributive property can be used for two different **arithmetic operations**, such as the following:

- Distributive Law in Multiplication
- Distributive Law in Division

**The Distributive Law in Multiplication**

One way to describe the distributive property of multiplication is by using **addition** and** subtraction operations**. That indicates that the operation is carried out within the **brackets**, specifically, **the addition or subtraction** that occurs between the **integers** within the brackets. Let’s understand these features better by looking at some examples.

**The Distributive Law of Multiplication Over Addition**

When you multiply a value, you use the **distributive property of multiplication**, which is an advantage that multiplication has over addition. As an illustration, you need to **multiply 6** by the **total of 5 and 3**.

Because these are related terms, we typically begin by **adding** the numbers together and then proceed to **multiply by 6:**

**6(5 + 3) = 6(8) = 48**

However, to fulfill the requirements of the** property**, you must first multiply each **addend by 6**. After completing this step, which is known as **dispersing the 6**, you may add the products.

Before you proceed with the addition, we will first perform the multiplication by **multiplying 6(5) by 6(3)**:

**6(5) + 6(3) = 30 + 18 = 48**

You’ll notice that the** outcome** is precisely the same as it was. You probably need to know that you are **employing** it in any given situation.

Both approaches are** modeled** by the equations that are presented below. On the **left-hand side**, we have the numbers **5 and 3**, which are then **multiplied by 6**. This expansion is rewritten by applying the distributive law on the **right-hand side**, where we **distribute 6**, **multiply by 6**, and then** sum the results**.

You will observe that the outcome is **comparable** in each of the three scenarios. The new form is as follows:

6(5 + 3) = 6(5) + 6(3)

6(8) = 30 + 18

**48 = 48**

**Distribution Law of Multiplication Over Subtraction**

Let’s look at an** illustration** of multiplication’s distributive property over the **arithmetic of subtraction**.

So, let’s say we need to **divide 8 and 5** from a **product of 6**.

There are **two methods** for achieving this result.

**Case 1:** 6 × (8 – 5) = 6 × 3 = 18

**Case 2:** 6 × (8 – 5) = (6 × 8) – (6 × 5) = 48 – 35 = 18

Whichever method is used; the outcome will be the same.

**Expressions** can be rewritten using the distributive properties of addition and subtraction. Using **addition and multiplication** is acceptable when multiplying by a sum. Another option is to multiply each addend separately and then add the resulting results. This also holds for **subtractions.** Multiplication occurs with each number inside the **parenthesis** before any addition or subtraction is performed because the outer **multiplier is shifted** in each case.

**What Is “The Distributive Law of Integers”?**

Integers have a feature known as the distributive property, which states that the product of an integer multiplied by the **sum of two integers** contained inside **parenthesis** is equivalent to the sum of the individual** products of integers**. Assuming that **a**, **b**, and **c** are all whole numbers, we may express the distributive property of multiplying integers by writing it as **[a (b + c)] = ab + bc**. This is because **integers’ multiplication** takes precedence over **integers’ addition**.

**Examples of the Distributive Law in Action**

**Example 1**

A dozen pencils cost \$3; a dozen pens cost \$5. Write an expression for purchasing four pencils and pens w.r.t the cost. Moreover, find the total cost using the distributive property.

**Solution**

The expression representing the pen and pencil cost:

4 x (Addition of cost of both the pencil and pen)

= 4 x (3 + 5)

The total cost becomes:

= (4 x 3) + (4 x 5)

= **$32**

**Example 2**

There are two boys and three girls in each row of a class, with a total of four rows. Find the total number of students in the class, considering the distributive property.

**Solution**

The expression representing the boys and girls in four rows:

4 x (Addition of cost of both the girls and boys in each row)

= 4 x (2 + 3)

The total students:

= (4 x 2) + (4 x 3)

= **20 Students**

*All mathematical drawings and images were created with GeoGebra.*