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Chapter 6: Problem 21

Simplify by removing a factor equal to 1. $$ \frac{2 a-10}{2} $$

Short Answer

Expert verified
\(a - 5\)

Step by step solution

01

Identify the Common Factor

Identify the common factor that can be factored out from the numerator. The numerator is \(2a - 10\). Both terms have a common factor of 2.
02

Factor Out the Common Factor

Factor the common factor of 2 from the numerator. \(2a - 10 = 2(a - 5)\)
03

Rewrite the Fraction

Substitute the factored form of the numerator back into the original fraction. \( \frac{2(a - 5)}{2} \)
04

Simplify the Fraction

Cancel the common factor in the numerator and the denominator. Since both are 2, they cancel each other out, resulting in \( a - 5 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is a key concept in algebra that involves breaking down an expression into simpler components. Think of it as reverse multiplication. Instead of multiplying terms to get a product, you are identifying terms that multiply together to reconstruct the original expression. For example, if you have the expression \(2a - 10\), you can factor out a common term, like 2, to get \(2(a - 5)\). This process makes the expression easier to work with in future steps, like in fraction simplification.
Common Factor
The common factor is the term that appears in all parts of an expression. Identifying common factors is critical for simplifying algebraic expressions. In the expression \(2a - 10\), the number 2 is a common factor for both terms (\(2a\) and -10). Once you identify the common factor, you can factor it out, meaning you divide each term by this common factor. This results in nested terms within a parenthesis, like \(2(a - 5)\). Finding the common factor makes the expression simpler and easier to manipulate.
Fraction Simplification
Simplifying fractions is about making them as simple as possible by cancelling out common factors in the numerator and denominator. For example, in the fraction \( \frac{2(a - 5)}{2} \), you can see that the number 2 is both in the numerator and the denominator. When they are the same, they cancel each other out. So, the simplified form of this fraction would be just \(a - 5\). This process is essential for solving equations and making expressions easier to understand.
Numerator and Denominator
In any fraction, the top part is called the numerator, and the bottom part is called the denominator. Understanding these terms is important for fraction simplification. In the example \( \frac{2(a - 5)}{2} \), \(2(a - 5)\) is the numerator, and 2 is the denominator. Both these parts can often have common factors that can be cancelled out to simplify the fraction. Knowing how to identify and manipulate the numerator and the denominator helps you manage and simplify complex algebraic expressions more effectively.

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