Problem 51 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{7}{a-4}+\frac{5}{4-a} $$

Step-by-Step Solution

Answer

\(\frac{2}{a-4}\)

Step 1: Identify the Denominators

Look at the denominators of the fractions. The two denominators are \(a-4\) and \(4-a\). Notice that \(4-a\) is the negation of \(a-4\). This means \(4-a = -(a-4)\).

Step 2: Equalize Denominators

To add these fractions, the denominators need to be identical. Since \(4-a = -(a-4)\), we can multiply the numerator and the denominator of the second fraction by \(-1\) to transform the denominator to \(a-4\). This gives us: \[ \frac{5}{4-a} = \frac{5 \cdot (-1)}{-(4-a)} = \frac{-5}{a-4} \]

Step 3: Add the Fractions

Now that both fractions have a common denominator of \(a-4\), they can be added together: \[ \frac{7}{a-4} + \frac{-5}{a-4} = \frac{7 + (-5)}{a-4} = \frac{2}{a-4} \]

Step 4: Simplify the Expression

The expression \(\frac{2}{a-4}\) is already in its simplest form because the numerator and denominator have no common factors other than 1. Therefore, no further simplification is possible.

Key Concepts

Understanding Common Denominators

When dealing with algebraic fractions, having common denominators is crucial when performing operations like addition or subtraction. A denominator is the bottom part of a fraction, and unifying them makes the operation straightforward. For the fractions \( \frac{7}{a-4} \) and \( \frac{5}{4-a} \), the denominators are different but related. Recognize that \( 4-a \) is the negative of \( a-4 \). This means that if you multiply the numerator and the denominator of \( \frac{5}{4-a} \) by \(-1\), the denominator becomes the same as \( a-4 \).

• Identify the denominators: \( a-4 \) and \( 4-a \).
• Transform \( 4-a \) to \( a-4 \) by multiplying by \(-1\).

By equalizing the denominators, we can now perform mathematical operations with ease.

Adding Algebraic Fractions

With a common denominator established, you can proceed to add algebraic fractions. The next step involves combining the numerators while keeping the common denominator unchanged.Consider the fractions \( \frac{7}{a-4} \) and \( \frac{-5}{a-4} \). Both share a denominator of \( a-4 \). Adding these fractions is as simple as:

• Add their numerators: 7 + (-5).
• Keep the denominator constant: \( a-4 \).

The result of adding these is \( \frac{7 + (-5)}{a-4} \), which simplifies to \( \frac{2}{a-4} \). This step-by-step method ensures clarity and accuracy when dealing with algebraic fractions.

Handling Negative Denominators

Negative denominators can confuse, but they are manageable with the right approach. Observe that a negative denominator can be rewritten in terms of a positive one by multiplying the fraction by \(-1\). For example, \( \frac{5}{4-a} \) has a denominator of \( 4-a \), which is the negative of \( a-4 \). By multiplying both the numerator and denominator by \(-1\), you achieve \( \frac{-5}{a-4} \):

• Understand that \( 4-a = -(a-4) \).
• Change the sign of both the numerator and denominator to match other denominators.

Using these strategies simplifies the process, allowing you to perform operations seamlessly across fractions with negative signs, ensuring consistent and correct results.