Problem 57 - Elementary and Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$ \frac{6(x+3)-18}{3 x-18} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{2x}{x-6} \).

Step 1: Simplify the Numerator

First, simplify the expression in the numerator, which is \( 6(x+3) - 18 \). Begin by distributing the \(6\) across the terms inside the parentheses:\[ 6(x+3) = 6x + 18. \]This gives us:\[ 6x + 18 - 18. \]Next, combine like terms \(18 - 18\), which simplifies to \(0\):\[ 6x. \]

Step 2: Simplify the Denominator

Now, simplify the denominator, which is \(3x - 18\). Notice that \(3x - 18\) has a common factor of \(3\):\[ 3x - 18 = 3(x - 6). \]

Step 3: Divide Both Numerator and Denominator by Their GCD

Now we have the simplified expression:\[ \frac{6x}{3(x-6)}. \]Divide both the numerator and the denominator by \(3\), their greatest common divisor (GCD):\[ \frac{6x}{3(x-6)} = \frac{2x}{x-6}. \]

Key Concepts

Understanding Like Terms

When working with algebraic expressions, like terms are crucial because they help us simplify expressions more efficiently. Like terms are terms that contain the same variables raised to the same powers. For example, in the expression \(6x + 3x\), both terms are like terms because they both contain the variable \(x\) to the first power. This allows us to combine them into \(9x\).

To simplify expressions by combining like terms, follow these steps:

• Identify terms that have identical variable parts with the same exponents.
• Add or subtract the coefficients of these terms while keeping the variable part unchanged.

It's a simple yet effective strategy that makes complex expressions easier to handle. Recognizing like terms is the first step towards mastering algebra.

Mastering the Distributive Property

The distributive property is a useful algebraic property that makes it possible to multiply a single term by each component within a set of parentheses. This is particularly useful when you have expressions like \(a(b + c)\), which can be expanded to \(ab + ac\).

Here's how you use the distributive property in a step-by-step manner:

• Identify the factor outside the parentheses (in this exercise, it's \(6\)).
• Multiply this factor by each term inside the parentheses.

In our example, \(6(x+3)\) becomes \(6x + 18\), which is simplified by distributing \(6\). This concept is fundamental as it aids in breaking down complex problems into simple arithmetic ones.

Practicing this method enhances your ability to solve equations quickly and accurately.

The Role of Greatest Common Divisor (GCD)

The greatest common divisor (GCD) plays a key role in simplifying fractions and expressions. The GCD of two or more terms is the largest factor that both terms share. Knowing how to find and use the GCD is essential for simplifying algebraic fractions.

To simplify an algebraic fraction using the GCD, follow these steps:

• Identify the GCD of the coefficients or terms in both the numerator and denominator.
• Divide both the numerator and the denominator by this GCD to simplify the fraction.

In the example provided, we found the GCD of \(6x\) (numerator) and \(3(x-6)\) (denominator) to be \(3\). By dividing both by this common factor, the expression simplifies to \(\frac{2x}{x-6}\).

This concept underscores the importance of factorization in algebra and helps make complicated expressions simpler, cleaner, and easier to interpret.