Problem 24 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{5 x-15}{9 x-27} $$

Step-by-Step Solution

Answer

\(\frac{5}{9}\)

Step 1: Factor the numerator

Factor out the greatest common factor (GCF) of the terms in the numerator. The numerator is \(5x - 15\), and the GCF is 5. So, \(5x - 15\) can be written as \(5(x - 3)\).

Step 2: Factor the denominator

Factor out the GCF of the terms in the denominator. The denominator is \(9x - 27\), and the GCF is 9. So, \(9x - 27\) can be written as \(9(x - 3)\).

Step 3: Simplify the fraction

Now the fraction is \(\frac{5(x - 3)}{9(x - 3)}\). Since \(x - 3\) is a common factor in both the numerator and the denominator, they cancel each other out. The simplified fraction is \(\frac{5}{9}\).

Key Concepts

greatest common factor

In algebra, finding the greatest common factor (GCF) of terms is crucial for simplifying expressions. The GCF is the largest number that divides all the terms completely. For example, in the numerator of our exercise, which is \(5x - 15\), the GCF is 5. This is because both 5 and 15 can be divided by 5.
To find the GCF, list the factors of each term and identify the highest common factor:

• Factors of 5x: 1, 5, x, 5x
• Factors of 15: 1, 3, 5, 15

(The factor we can see here is 5)
We use this factor to rewrite the expression by factoring it out, giving us \(5(x - 3)\). Similarly, for the denominator \(9x - 27\), the GCF is 9, which means we factor it like this: \(9(x - 3)\).
Extracting the GCF simplifies the process of breaking down and simplifying fractions.

factoring expressions

Factoring expressions involves breaking down complex terms into simpler factors. This makes it easier to simplify and solve equations.
In our example, we need to factor both the numerator and the denominator.

• For the numerator \(5x - 15\), we factor out the greatest common factor (GCF), which is 5, resulting in \(5(x - 3)\).
• For the denominator \(9x - 27\), we factor out the GCF, 9, resulting in \(9(x - 3)\).

Now, the fraction is:\( \frac{5(x - 3)}{9(x - 3)} \). Recognize the repeated factor, \(x - 3\), in both terms.
Factoring brings terms into a form that makes further simplification and cancellations possible.

simplification

Simplification is about reducing expressions to their simplest form. It's the last step where we want to make everything as clean and concise as possible.

In our exercise, after factoring both the numerator and the denominator, we get: \(\frac{5(x - 3)}{9(x - 3)}\). Notice that \(x - 3\) appears in both the numerator and the denominator. These terms cancel each other out, leaving us with the simplified fraction \( \frac{5}{9} \).
The cancellation works because dividing any number by itself equals 1 (\( \frac{(x - 3)}{(x - 3)} = 1 \)). Keep in mind:

• Always factor expressions entirely.
• Identify and cancel common factors in fractions.

Simplifying makes expressions easier to understand and work with.