Problem 19 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{(y-1)(y-7)}{(y-1)(y+6)} $$

Step-by-Step Solution

Answer

Question: Reduce the rational expression to its lowest terms: $\frac{(y-7)(y-1)}{(y-1)(y+6)}$ Answer: $\frac{(y-7)}{(y+6)}$

Step 1: Identify any common factors in the numerator and denominator

In this case, we see that both the numerator and the denominator have the factor \((y-1)\).

Step 2: Cancel out the common factors

To cancel out the common factor \((y-1)\), we'll divide both the numerator and the denominator by \((y-1)\). We're left with: $$ \frac{(y-7)}{(y+6)} $$

Step 3: Check for any more common factors

There are no more common factors in the numerator and the denominator. Thus, the rational expression has been reduced to its lowest terms: $$ \frac{(y-7)}{(y+6)} $$

Key Concepts

Understanding the Numerator and Denominator

In any rational expression, you will encounter two main parts: the **numerator** and the **denominator**. The numerator is the expression located above the fraction line, while the denominator is below it. In the problem \( \frac{(y-1)(y-7)}{(y-1)(y+6)} \), the numerator is \((y-1)(y-7)\) and the denominator is \((y-1)(y+6)\).
These components are critical because they form the fraction, or rational expression, that needs to be simplified. Both the numerator and denominator can contain multiple terms and factors, and understanding them helps in identifying any potential common factors to simplify the expression.
When tackling problems involving rational expressions, always identify and separate the numerator from the denominator. This clarity will guide you through the simplification process and make it easier to eliminate unnecessary factors.

Identifying Common Factors

Finding common factors is a crucial step in simplifying a rational expression. A **common factor** refers to any term or number that appears in both the numerator and the denominator of the expression.
For our problem, \( \frac{(y-1)(y-7)}{(y-1)(y+6)} \), we observe that the term \((y-1)\) is present in both the numerator and the denominator. Identifying this similarity is essential because these common factors are the candidates that can be canceled out.
To better identify common factors, look for:

• Repeating terms or numbers in both parts of the expression
• Factors that can be grouped or rearranged to reveal hidden common factors

Once you've recognized common factors, you can simplify the expression significantly by canceling them out.

The Process of Cancelling Factors

Cancelling factors is the mechanism that allows us to simplify rational expressions. After identifying common factors in both the numerator and the denominator, the next step is to "cancel" or eliminate these factors.
This is done by dividing both the numerator and the denominator by the common factor. In our example, by dividing by \((y-1)\), we are left with \( \frac{(y-7)}{(y+6)} \). In essence, cancelling means reducing the expression to its simplest equivalent form.
It's important to ensure that:

• The factor truly appears in both the numerator and denominator
• All common factors are cancelled out to reach the simplest form
• We do not cancel out terms that are not actual factors

This simplification makes the rational expression easier to work with and understand, especially when solving further algebraic problems.