Problem 5 - Algebra A Combined Function

Question

$$ \frac{4 m}{m-6}-\frac{24}{m-6} $$

Step-by-Step Solution

Answer

The simplified expression is 4.

Step 1: Identify Common Denominator

Look at the given expression: \( \frac{4m}{m-6} - \frac{24}{m-6} \). Notice that both fractions have the same denominator, \( m-6 \). This means we can directly combine the two fractions into a single fraction.

Step 2: Combine the Numerators

Since the denominators are the same, we can subtract the numerators: \( 4m - 24 \). The combined expression is: \( \frac{4m - 24}{m-6} \).

Step 3: Simplify the Numerator

Factor the numerator if possible. The expression \( 4m - 24 \) can be factored as \( 4(m-6) \).

Step 4: Simplify the Fraction

Now, substitute the factored numerator back into the fraction: \( \frac{4(m-6)}{m-6} \). Since \( m-6 \) is in both the numerator and the denominator, we can cancel these out.

Step 5: Final Result

After canceling \( m-6 \) from the numerator and denominator, we are left with \( 4 \). Thus, the simplified expression is \( 4 \).

Key Concepts

Common Denominator

A common denominator is a fundamental concept in fraction calculations. In equations involving fractions, having the same denominator simplifies the operation, making it straightforward to add or subtract them.

• In the given exercise, we see two fractions: \( \frac{4m}{m-6} \) and \( \frac{24}{m-6} \).
• Both fractions possess the denominator \( m-6 \), which indicates that they already have a common denominator.

This means we can combine them easily by focusing only on their numerators without dealing with the denominators anymore.

Numerator Subtraction

Numerator subtraction is the process of subtracting the numerators of two fractions that share a common denominator.

• With the same denominator \( m-6 \), we directly subtract the numerators of the fractions: \( 4m - 24 \).
• The result gives us a new fraction: \( \frac{4m - 24}{m-6} \).

The subtraction here simplifies the transition into a single, unified expression. It's crucial for preparing the fraction for further simplification steps.

Factoring Expressions

Factoring is a technique used to simplify expressions by finding elements common across the terms.

• In the expression \( 4m - 24 \), we notice that both terms can be divided by \( 4 \).
• This allows us to rewrite \( 4m - 24 \) as \( 4(m-6) \), bringing the expression into a factorable form.

By factoring, we break down the expression into simpler components. It particularly helps when simplifying fractions, setting up the stage for potential term cancellation.

Canceling Terms

Canceling terms is an important step in simplifying fractions. It involves removing or reducing parts of the expression that appear in both the numerator and the denominator.

• Once we factor the numerator to \( 4(m-6) \), we see that \( m-6 \) appears in both the numerator and denominator.
• We can cancel out \( m-6 \), provided it’s not equal to zero.
• This leaves us with the simple expression: \( 4 \).

Canceling here effectively removes complexity, giving us the final simplified form of the fraction. It demonstrates how the combination of factoring and finding identical terms can lead to significant simplification.