Problem 59 - Algebra 2

Question

Name the property illustrated by each equation. \(m(4-3)=m \cdot 4-m \cdot 3\)

Step-by-Step Solution

Answer

Distributive Property.

Step 1: Identify the Operation Type

The equation provided involves operations that are combined under parentheses and then distributed outside the parentheses. It is important to recognize that the structure is an example of a multiplication operation affecting terms within parentheses.

Step 2: Recognize the Equation Structure

The equation given is \( m(4-3) = m \cdot 4 - m \cdot 3 \). This structure is commonly used when a single term outside a set of parentheses multiplies each term within the parentheses separately.

Step 3: Associate the Structure with a Property

The property illustrated by this equation is known as the Distributive Property. The property states that multiplying a sum (or difference) by a factor is the same as multiplying each addend (or subtrahend) separately by the factor and then adding (or subtracting) the products.

Key Concepts

Multiplication

Multiplication is an essential arithmetic operation, foundational to various mathematical concepts. At its core, multiplication is a way of adding a number multiple times. For instance, multiplying 3 by 4 means you add 3, four times: 3 + 3 + 3 + 3 = 12.
In the context of the distributive property, multiplication becomes even more powerful. It allows us to take a number outside of parentheses and multiply it by each term inside the parentheses separately. This strategy is especially useful in breaking down complex equations and simplifying calculations.
For example, in the equation \( m(4-3) \), the multiplication is distributed as \( m \cdot 4 \) and \( m \cdot 3 \). This helps us see clearly how multiplication interacts with other operations, making problem-solving more efficient.

Equation Structure

The structure of an equation is crucial in understanding how different mathematical operations connect. An equation provides a balanced relationship between its two sides, allowing us to manipulate and solve it.
In the equation \( m(4-3) = m \cdot 4 - m \cdot 3 \), the structure showcases how distributive multiplication operates within an equation. The original form, \( m(4-3) \), indicates a single multiplication operation applied to a group of terms. When rewritten as \( m \cdot 4 - m \cdot 3 \), the structure explicitly shows that each term inside the parentheses is individually multiplied by \( m \).
This structure not only helps to break down the elements within the equation but also assists in identifying properties like the distributive property that govern these operations.

Properties of Operations

Properties of operations are rules that define how arithmetic operations interact within expressions and equations. Understanding these properties is fundamental in simplifying expressions and solving equations effectively.
The Distributive Property is one of the key properties of operations. It states that when you multiply a number by a sum or difference, it's equivalent to multiplying the number by each term of the sum or difference, then performing the operation with those products. This means \( a(b + c) = ab + ac \) and similarly, \( a(b - c) = ab - ac \).
This property is practical in mathematics as it simplifies expressions and eases the process of equation solving. It plays a pivotal role in algebra and higher-level mathematics, providing a strategy to manage complex expressions by breaking them into more manageable parts.