Problem 88 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. Divide and simplify: \(\frac{-7 y}{-7}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{-7 y}{-7}\) is 'y'.

Step 1: Identify the Numerator and Denominator

In the given expression \(\frac{-7 y}{-7}\), '-7y' is the numerator and '-7' is the denominator.

Step 2: Simplify the Fraction

To simplify, the -7 in the numerator is divided by the -7 in the denominator. The minus signs cancel each other out, leaving you with \(\frac{y}{1}\).

Step 3: Final Simplification

Any number divided by 1 equals the number itself. Therefore, \(\frac{y}{1}\) simply equals 'y'.

Key Concepts

Numerators and Denominators

Understanding the function of numerators and denominators is essential in fraction operations. In a fraction, the number on top is the numerator and it represents how many parts we have. The number on the bottom is the denominator, which tells us into how many parts the whole is divided. For instance, in the fraction \(\frac{-7y}{-7}\), the -7y is the numerator, indicating the number of parts, while the -7 is the denominator, representing the divisions of a whole.

In simplifying fractions, our goal is to reduce both numerator and denominator to their smallest possible values while keeping the fraction equivalent to its original form. When both numerator and denominator share a common factor, in this case, the number \(7\), we can divide both by that common factor to simplify the fraction.

Simplifying Expressions

The process of simplifying expressions involves reducing them to their most basic form without changing their value. In algebra, this could mean combining like terms, reducing fractions, or factoring. When we simplify \(\frac{-7y}{-7}\), we see that the common factor is \(7\), and specifically, because they are the same number with opposite signs, they cancel each other out, leading to \(\frac{y}{1}\).
It's also important to look for signs during simplification. Since both the numerator and denominator are negative, they cancel each other, turning our expression positive. This is a fundamental rule in mathematics—they are what we call inverse elements with respect to multiplication, as their product is one, the multiplicative identity.

Algebraic Fractions

When dealing with algebraic fractions, the principles of simplifying numbers apply with the additional consideration of variables. In the fraction \(\frac{-7y}{-7}\), we see algebra at play through the variable \(y\). After simplification, we're left with \(\frac{y}{1}\) or just \(y\), which is an algebraic expression representing an unspecified or variable quantity. It's essential to recognize that once the numerical coefficients are simplified, variables remain intact unless they are part of the simplification process.

By mastering algebraic fractions, you gain the ability to work with variables in the same way you work with numbers. When dividing variables with like bases, you also subtract exponents according to the laws of exponents. This concept is a cornerstone for higher-level math and is frequently used in solving equations and inequalities.