Problem 66 - Algebra 1

Question

Simplify the expression. $$ \frac{6 x+12 y}{24} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{6 x+12 y}{24}\) is \(\frac{x+2y}{4}\).

Step 1: Factor out common factors

Look at the numerator of the fraction. Here, both terms in the numerator, \(6x\) and \(12y\), have a common factor of 6. So we can factor that out to get \(6( x+2 y)\). After factoring out, our expression becomes \(\frac{6(x+2y)}{24}\).

Step 2: Simplify the expression

Now look at the entire fraction. In this step we notice that both the numerator and the denominator have a common factor, which is 6. Divide both the numerator and the denominator by this common factor. The expression simplifies to \(\frac{x+2y}{4}\).

Key Concepts

Factoring

Factoring is a crucial algebraic method for simplifying expressions. At its core, factoring involves finding numbers or expressions that multiply together to form the original expression. When simplifying, the goal is to express a given term as a product of its factors. For instance, in the expression \(6x + 12y\), both terms \(6x\) and \(12y\) have a common numerical factor of 6.

• Identify the greatest common factor (GCF) in the terms of the numerator.
• In this example, it's 6, which means 6 can be multiplied by another expression to form each term in the numerator.
• Rewrite the numerator as \(6(x + 2y)\).

Factoring simplifies the terms and potentially sets the stage for canceling out terms with the denominator if similar factors exist.

Common Factors

Common factors are factors that appear in more than one term or expression. In our example, both \(6x + 12y\) in the numerator and the number 24 in the denominator share a common factor of 6.

• To identify common factors, break down each term into its prime or simplest components.
• Find numbers that appear in all these breakdowns.
• In our fraction, the common factor 6 was used to simplify the expression by division.

By spotting and using common factors, expressions can be reduced to simpler forms, increasing ease of calculation and insight into the structure of the expression.

Numerators and Denominators

The numerator is the top part of a fraction, while the denominator is the bottom part. Simplifying a fraction involves finding common factors in both to streamline the expression. Let's break down our fraction \(\frac{6(x+2y)}{24}\).

• The numerator \(6(x + 2y)\) shows each part of the original top term is divided by 6.
• The denominator 24 also shares this factor of 6.
• By dividing both by 6, we reduce the fraction to \(\frac{x+2y}{4}\).

Understanding numerators and denominators and their relationship helps clarify fraction simplification. It is about focusing on shared factors and reducing the terms to their simplest forms, without changing the value of the fraction itself.