Problem 87 - Elementary Algebra Within Reach

Question

In Exercises 85-88, identify the property of algebra illustrated by the statement. $$ 2(c-d)=2 c-2 d $$

Step-by-Step Solution

Answer

The given equation \( 2(c-d)=2c-2d \) illustrates the Distributive Property.

Step 1: Understanding the given equation

First, it is noted that the equation is \( 2(c-d)=2c-2d \). This indicates some kind of simplification has happened, since the expression \( c-d \) has been multiplied by 2 on the left side of the equation, and on the right side of the equation, both \( c \) and \( d \) are individually multiplied by 2.

Step 2: Identify the algebraic property

The equation shows that multiplication is distributive over subtraction within \( c-d \). This means that multiplying \( 2(c-d) \) is the same as multiplying 2 with \( c \) and \( d \) individually and then subtracting these values. Therefore, the given equation illustrates the Distributive Property of Algebra.

Key Concepts

Algebraic Properties

Understanding the fundamental algebraic properties is crucial for simplifying and manipulating expressions. The distributive property, as illustrated in the exercise, is one of the most valuable algebraic properties. It allows you to distribute a multiplied number across terms within parentheses.

In general, the distributive property states that for any numbers a, b, and c, the equation a(b + c) = ab + ac holds true. Similarly, it works with subtraction: a(b - c) = ab - ac. This property is essential not only in elementary algebra but also in higher mathematics where expressions become more complex.

Simplifying Expressions

Simplifying expressions is a process of reducing them to their simplest form, making them easier to work with. This often involves applying the distributive property, combining like terms, and performing arithmetic operations.

In the provided exercise, simplification involves applying the distributive property to eliminate the parentheses. Breaking down an expression into simpler parts can help in solving various algebra problems. The goal is to express the mathematics in as few terms as possible without changing the original value.

Multiplication Over Subtraction

Multiplication over subtraction is a specific application of the distributive property. It occurs when a number is multiplied by a difference.

For example, the expression 2(c - d) is dealing with the multiplication of 2 over the subtraction of d from c. By distributing the 2, we see that each term within the parentheses is multiplied by 2, resulting in 2c - 2d, demonstrating that the subtraction inside does not interfere with the multiplication outside. Understanding this process is quite helpful, especially as equations become more involved.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with solving equations involving variables, like the exercise provided. It introduces the concept of using symbols and letters to represent numbers and the relationships between them.

The distributive property is just one of many building blocks in elementary algebra. Mastery of these principles forms a foundation that supports all further studies in mathematics, particularly in algebra. Students are encouraged to get comfortable with these concepts early on as they form the basis for more complex problem-solving strategies in advanced algebra.