Problem 7 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{15}{x-9}-\frac{6}{x-9} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{9}{x-9}\).

Step 1: Identify the Common Denominator

Observe that both terms, \(\frac{15}{x-9}\) and \(\frac{6}{x-9}\), have the same denominator of \(x-9\).

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators directly: \(\frac{15 - 6}{x-9}\).

Step 3: Simplify the Fraction

Calculate the difference in the numerator: \(15 - 6 = 9\). Thus, the simplified form of the expression is \(\frac{9}{x-9}\).

Key Concepts

Common Denominator

When working with fractions, finding a common denominator is essential to perform operations like addition or subtraction.
The denominator is the bottom part of a fraction, and a common denominator means both fractions are expressed in terms of the same base number.
In the given exercise, the fractions \(\frac{15}{x-9}\) and \(\frac{6}{x-9}\) already have a common denominator, which is \(x-9\).
This common denominator allows us to combine the fractions seamlessly. Without a common denominator, it's impossible to directly add or subtract fractions.

Subtracting Fractions

Once a common denominator is identified, subtracting fractions becomes much easier.
Here’s a simple rule: keep the common denominator and operate only on the numerators.
So, if we need to subtract \(\frac{6}{x-9}\) from \(\frac{15}{x-9}\), we simply subtract the numbers on top (the numerators).
The operation looks like this: \(\frac{15 - 6}{x-9}\).
Notice how the denominator \(x-9\) remains unchanged during this process.

Simplifying Fractions

After subtracting the numerators, the next step is simplifying the fraction.
To simplify means to reduce the fraction to its lowest terms.
In this case, after performing the subtraction in the numerator, we get \(15 - 6 = 9\).
This results in the fraction \(\frac{9}{x-9}\).
Verify if the numerator and the denominator have any common factors other than 1.
If they do, divide by the greatest common factor to simplify further.
In our example, \(\frac{9}{x-9}\) is already in its simplest form, assuming \(x\) is not equal to 9 (to avoid division by zero).
Simplifying fractions makes them easier to understand and work with in further mathematical tasks.