Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the rule of algebra illustrated by the statement. \(7\left(\frac{1}{7}\right)=1\)

Step-by-Step Solution

Answer

The algebra rule illustrated by the given statement is the Multiplicative or Reciprocal Identity, which states that a number multiplied by its reciprocal equals 1.

Step 1: Identify the Numbers

Firstly, identify the numbers involved in the exercise. Here, 7 and \(\frac{1}{7}\) are the numbers involved.

Step 2: Understand the Reciprocal Relationship

Next, understand the relationship between the two numbers. Here, they are reciprocals as the product of a number and its reciprocal is always 1.

Step 3: Identify the Algebra Rule

Finally, recognize that this is a demonstration of the rule in algebra that states that a number multiplied by its reciprocal always equals 1. This is known as the Multiplicative Identity or the Reciprocal Identity.

Key Concepts

Algebraic Rules

The foundation of algebra rests on a set of rules that dictate how numbers and variables interact. Among these are the distributive, associative, and commutative properties, but one rule that is integral to equation solving and simplifying expressions is the Multiplicative Identity.

In algebra, the Multiplicative Identity is a principle which states that any number multiplied by one retains its original value. It is one of the cornerstones of arithmetic and serves as a basic tool in equation solving. When applied, it ensures the number remains unchanged, thereby maintaining the integrity of the operation. To put it simply, if you multiply any number by 1, the result will be the number itself.

Reciprocal Numbers

At the heart of the concept of multiplicative identity lies the idea of reciprocal numbers. A reciprocal of a number is defined as the value which, when multiplied with the original number, results in the multiplicative identity of 1. For any non-zero number 'a', its reciprocal is represented as \(\frac{1}{a}\).

Understanding reciprocals is crucial, especially when dealing with fractions and division. For example, the reciprocal of 2 is \(\frac{1}{2}\), and for \(\frac{3}{4}\), it is \(\frac{4}{3}\). When reciprocals are used in multiplication, they 'cancel out' each other, resulting in 1. This plays a significant role in simplifying algebraic expressions and solving equations.

Multiplication of Reciprocals

Multiplication of reciprocals is the process where a number is multiplied by its reciprocal, leading to the product of 1. This is sometimes referred to as the 'Inverse Property of Multiplication', where the reciprocal acts as the inverse element.

The importance of this operation comes into play when you're working with equations and need to isolate a variable. By multiplying both sides of an equation by the reciprocal of a coefficient, you can effectively 'get rid' of the coefficient and solve for the variable. For instance, if you have an equation such as \(7x = 14\), you can multiply both sides by the reciprocal of 7, which is \(\frac{1}{7}\), to obtain the solution \(x = 2\). It's a key maneuver that enables us to untangle variables from their coefficients.