Problem 64 - Algebra 2

Question

Name the property illustrated by each equation. $$ \frac{7}{9} \cdot \frac{9}{7}=1 $$

Step-by-Step Solution

Answer

The equation illustrates the Multiplicative Inverse Property.

Step 1: Identify the Equation Type

Examine the given equation: \( \frac{7}{9} \cdot \frac{9}{7}=1 \). Notice that two fractions are being multiplied together and the result is 1.

Step 2: Recognize the Multiplicative Inverse

Identify that for two numbers \, \(\textit{a}\) \, and \, \(\textit{b}\) \, to satisfy the equation \( a \cdot b = 1 \), each must be the multiplicative inverse of the other. Here, \( \frac{7}{9} \) and \( \frac{9}{7} \) are multiplicative inverses because if you multiply them, the result is 1.

Step 3: State the Given Property

Understand that the property illustrated by \( \frac{7}{9} \cdot \frac{9}{7}=1 \) is the Multiplicative Inverse Property. This property states that any number multiplied by its reciprocal (inverse) equals 1.

Key Concepts

Reciprocal

In mathematics, a reciprocal is a unique number that when multiplied with the original number results in the product of one. This idea comes into play particularly with fractions. For a given fraction, say \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \). When these two are multiplied together, they cancel each other out, yielding one. This is because multiplying both the numerator and the denominator results in the use of the whole number one, ensuring the concept of inverses in multiplication.

Understanding the concept of reciprocals is crucial when solving equations involving the Multiplicative Inverse Property. A reciprocal not only helps you verify multiplication correctness, but it also streamlines simplifications in algebraic expressions, especially when fractions are involved. Always remember, the reciprocal is like a mirror reflection; it flips the numbers around the division line, but the multiplied result always brings us back to one.

Properties of Multiplication

The Properties of Multiplication help us understand how numbers interact under multiplication. They form a foundational part of arithmetic and algebra. Here are a few vital properties:

• *Multiplicative Identity*: This property states that any number multiplied by one remains unchanged. Formally, \( a \times 1 = a \).


• *Commutative Property*: It tells us that changing the order of numbers in multiplication does not affect the result, for instance, \( a \times b = b \times a \).


• *Associative Property*: It reveals that when three numbers are multiplied, the grouping does not affect the product, i.e., \( (a \times b) \times c = a \times (b \times c) \).


• *Distributive Property*: Offers a way to multiply a number by a sum, expressed as \( a \times (b + c) = a \times b + a \times c \).

The Multiplicative Inverse Property is another essential element, indicating how a number finds its pair, the reciprocal, resulting in one when multiplied. This highlights how deeply interconnected multiplication's properties can be, providing pathways for simplifying and solving mathematical problems effectively.

Inverse Operations

Inverse operations are actions that undo each other, working like a key and lock. When we talk about inverse operations in multiplication, we often think about the relationship of a number with its reciprocal. A number times its inverse will result in one, perfectly demonstrating the Multiplicative Inverse Property. Imagine you wish to balance a scale; the inverse operation will ensure both sides remain equal.

This concept becomes handy in simplifying algebraic equations and performing checks. For instance, knowing the inverse of a fraction helps quickly resolve division into multiplication, as dividing by a fraction is the same as multiplying by its inverse. Practicing inverse operations sharpens one's ability to manipulate and solve equations efficiently, reinforcing the significance of concepts like reciprocals in everyday mathematics.