Problem 87 - Elementary Algebra

Question

Use this approach to help with the additions and subtractions. $$\frac{-4}{a-1}+\frac{2}{1-a}$$

Step-by-Step Solution

Answer

\(\frac{-6}{a-1}\)

Step 1: Identify Like Terms

First, notice that both fractions have the same denominator but with opposite signs. The denominators are \(a-1\) and \(1-a\). Realize that \(1-a = -(a-1)\).

Step 2: Simplify the Second Fraction

Rewrite the second fraction \(\frac{2}{1-a}\) as \(-\frac{2}{a-1}\) because \(\frac{2}{1-a}\) is equivalent to \(-\frac{2}{a-1}\).

Step 3: Combine the Fractions

Now that both fractions have the same denominator \(a-1\), combine them: \[ \frac{-4}{a-1} + \left(-\frac{2}{a-1}\right) = \frac{-4 - 2}{a-1}. \]

Step 4: Simplify the Numerator

Simplify the numerator by performing the subtraction: \[ -4 - 2 = -6. \]

Step 5: Write the Final Expression

Express the combined fraction with the simplified numerator: \[ \frac{-6}{a-1}. \]

Key Concepts

Addition and Subtraction of Fractions

When working with fractions, it is crucial to have a common denominator before adding or subtracting them. The denominator is the number below the fraction line. If fractions have different denominators, you must first find equivalent fractions with a common denominator.
A simple trick when dealing with negative signs is to notice that changing the order of subtraction can lead to compatible denominators. Such was the case in our original exercise. We were given fractions with denominators in opposite signs. Recognizing that \(1-a = -(a-1)\) allowed us to rewrite \(\frac{2}{1-a}\) as \(-\frac{2}{a-1}\).
This approach helps in simplifying and managing fractions where the denominators are just negative versions of each other, making the addition or subtraction much more straightforward. Once both fractions share the same denominator, you can easily add or subtract them by combining the numerators over this common denominator.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This makes the expressions easier to understand and work with.
Key methods for simplification include:

• Combining like terms (terms that have the same variable raised to the same power).
• Factoring expressions to find common factors.
• Canceling out terms that appear in both the numerator and denominator.

During the exercise, we made a crucial step that significantly simplified our expression. By rewriting \(\frac{2}{1-a}\) to \(-\frac{2}{a-1}\), we instantly allowed the two fractions to have a common denominator. This enabled us to combine them without additional steps to find or adjust any denominator further. Once you reached a point where all possible like terms are combined, and all factors have been considered, the expression is in its simplest possible form.

Like Terms in Algebra

In algebra, the concept of like terms is pivotal for simplifying expressions and solving equations effectively. Like terms are terms whose variables (and their exponents) are the same. These terms can be added or subtracted as if they were simple numerical terms.
For instance, in the process of our exercise, we were able to recognize and manage like terms effectively. The terms \(-4\) and \(-2\) were both in the numerators of the fractions with a common denominator of \(a-1\). This allowed us to directly add them together, just as with regular numbers: \(-4 - 2 = -6\).
To quickly identify like terms, look for:

• Consistent variable names
• The same power for each variable in the terms

Adding or subtracting like terms simplifies expressions, leading to cleaner and easily manageable results. They are a critical component of algebra that aids not only in expression manipulation but also in solving equations efficiently.