Problem 16 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{5}{a-7}+\frac{5}{7-a}\)

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Identify Common Terms

Observe that the denominators \(a-7\) and \(7-a\) can be rewritten as each other's negatives. Specifically, \(7-a = -(a-7)\).

Step 2: Rewriting the Second Fraction

Rewrite the second fraction by factoring out a negative sign from its denominator: \[ \frac{5}{7-a} = \frac{5}{-(a-7)} = -\frac{5}{a-7} \]

Step 3: Combine the Fractions

Since both fractions now have the same denominator \((a-7)\), they can be combined: \[ \frac{5}{a-7} - \frac{5}{a-7} = \frac{5-5}{a-7} \]

Step 4: Simplify the Expression

Simplify the expression obtained in Step 3: \[ \frac{5-5}{a-7} = \frac{0}{a-7} = 0 \]

Key Concepts

Understanding Denominators

In the world of algebra, the denominator is a crucial component of any fraction. It is the bottom part of the fraction that tells you how many equal parts the whole is divided into. For example, in the fraction \( \frac{5}{a-7} \), \(a-7\) is the denominator. It specifies the portion of the whole that you are dealing with.

When adding or subtracting fractions, having a common denominator is essential. This allows the fractions to be combined directly. In our problem, the original fractions are \( \frac{5}{a-7} \) and \( \frac{5}{7-a} \). Notice that the denominators look different but are actually negatives of each other.

To proceed with operations, it is crucial to rewrite them so they represent the same number. This involves factoring negatives, which we'll touch upon next.

The Role of Factoring

Factoring is a fundamental skill that makes working with algebraic fractions much easier. It involves rewriting an expression as a product of its factors.

In the context of our exercise, factoring helps us simplify denominators. The fraction \( \frac{5}{7-a} \) is rewritten by factoring out a negative sign, hence:

• Identify that \(7-a\) is the same as \(-(a-7)\).
• Rewrite the expression as \(-\frac{5}{a-7}\).

Factoring like this transforms both fractions to have a common denominator \((a-7)\), making it easier to perform arithmetic operations. Understanding how to factor expressions is vital as it enables simplification and paves the way for combining fractions efficiently.

Simplifying Algebraic Expressions

Simplifying expressions is about reducing them to their most concise form. It's a key skill that allows you to work with expressions effortlessly.

In our example, once the fractions \( \frac{5}{a-7} \) and \(-\frac{5}{a-7} \) have a common denominator, they can be combined:

• Add or subtract the numerators while keeping the denominator the same.
• This results in: \( \frac{5-5}{a-7} \).

The numerator simplifies to zero, leading to \( \frac{0}{a-7} = 0 \). Simplifying in this way eliminates unnecessary terms and clarifies the solution. By mastering simplification, you enhance your ability to solve more complex algebraic problems effectively.