Problem 43 - Elementary and Intermediate Algebra

Question

Simplify. See Example 4. $$ \frac{6 x+3}{9} $$

Step-by-Step Solution

Answer

\( \frac{2x + 1}{3} \)

Step 1: Identify the Numerator and Denominator

In the expression \( \frac{6x + 3}{9} \), identify the numerator as \( 6x + 3 \) and the denominator as \( 9 \).

Step 2: Factor the Numerator

Look for a common factor in the numerator \( 6x + 3 \). The greatest common factor (GCF) of \( 6x \) and \( 3 \) is \( 3 \). Factor out \( 3 \): \[ 6x + 3 = 3(2x + 1) \].

Step 3: Simplify the Fraction

Rewrite the expression with the factored numerator: \[ \frac{3(2x + 1)}{9} \]. Simplify by dividing the numerator and the denominator by their greatest common factor, which is \( 3 \): \[ \frac{3(2x + 1)}{9} = \frac{2x + 1}{3} \].

Step 4: Final Simplified Expression

The simplification process results in the expression \( \frac{2x + 1}{3} \). This is the simplest form of the given expression.

Key Concepts

Factoring in Algebra

Factoring is a fundamental concept in algebra that involves breaking down a complex expression into simpler 'factors'. Think of it as the opposite of distributing or expanding an expression. In the exercise, we had the expression \( 6x + 3 \) in the numerator. By identifying common factors, we can simplify this expression.

The goal of factoring is to find a number or variable that evenly divides each term in the expression. This common number or variable is called a 'factor'. Many times when factoring, students look for the greatest common factor (GCF).

For example, in \( 6x + 3 \), the terms can both be divided by the number 3. So, 3 is a factor. When we factor 3 out, the expression becomes \( 3(2x + 1) \). By recognizing and factoring common elements, we streamline the simplification process, enabling us to deal with less complex fractions.

Greatest Common Factor

The Greatest Common Factor (GCF) is an important element in algebra that involves finding the largest number that divides each term in an expression without leaving a remainder. Determining the GCF is crucial for simplifying expressions, especially fractions.

For the expression \( 6x + 3 \), the GCF is 3. We find it by identifying the prime factors of each term. The term \( 6x \) consists of factors \( 2, 3, \) and \( x \), and 3 has just the factor 3. The greatest factor they share is 3.

Using the GCF, we can rewrite the numerator of the fraction as a product. By factoring the GCF out of \( 6x + 3 \), it simplifies to \( 3(2x + 1) \). A simplified fraction is easier to understand and can make further calculations more manageable, especially in larger algebra problems.

Fractions in Algebra

Fractions in algebra serve the purpose of dividing expressions into manageable parts. Unlike simple numeric fractions, algebraic fractions often involve variables and require additional steps to simplify.

Consider the expression \( \frac{6x + 3}{9} \). The initial step is to look for common factors in the numerator and denominator. After factoring \( 3(2x + 1) \) from the numerator, both the numerator and the denominator have a common factor of 3.

To simplify the fraction, we can divide both the top and bottom by the GCF, which in this case is 3. Thus, it simplifies the expression to \( \frac{2x + 1}{3} \). This simplification reduces complexity and reveals opportunities for further evaluation or computation.
Simplifying fractions helps in clarifying what an expression truly represents and in performing operations like addition or multiplication more straightforwardly.