Problem 67 - Introductory Algebra for College Students

Question

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{2 y}{y-5}-\left(\frac{2}{y-5}+\frac{y-2}{y-5}\right)$$

Step-by-Step Solution

Answer

The result of performing the indicated operation and simplifying the given expression will be \(y / (y - 5)\).

Step 1: Understand the Problem

The problem is asking to subtract two fractions where both have the same denominators. The first fraction is \(2y/(y-5)\) and the second one is a sum of two fractions which can be combined since they have the same denominator.

Step 2: Simplify the Expression Inside the Parentheses

The expression inside the parentheses can be rewritten as \((2 + y - 2) / (y - 5)\) which simplifies to \(y / (y - 5)\). Thus, the entire expression now reads \(2y/(y-5) - y/(y-5)\).

Step 3: Perform the Subtraction

Since both fractions have the same denominator, the subtraction can be performed by subtracting the numerators, which gives the result \( (2y - y) / (y - 5) = y / (y - 5)\).

Key Concepts

Subtracting Fractions

Working with fractions often involves addition or subtraction, and in both cases, having a common denominator simplifies the process. When subtracting fractions, you need to ensure that the denominators are the same. This is key because fractions represent parts of a whole, and the denominators indicate the size of those parts.
When denominators match, you simply subtract the numerators. For example, in the expression \( \frac{2y}{y-5} - \frac{y}{y-5} \), the denominator \( y-5 \) is already common to both terms. So, you need to only subtract the numerators:

• Subtract \( y \) from \( 2y \), giving you \( 2y - y = y \).

Thus, the subtraction simplifies to \( \frac{y}{y-5} \). Simplicity and clarity in numerators make subtraction straightforward when denominators match.

Simplifying Expressions

Simplifying expressions makes them easier to work with, reveal solutions more clearly, and illustrate relationships between terms. It's essential for solving equations efficiently. To simplify an algebraic expression, break it down and combine like terms. Looking at our example, after subtracting fractions, we are left with \( \frac{y}{y-5} \). This is already simplified because it has a single term in the numerator and no terms that cancel in the denominator.
Sometimes, you might need to factor terms out or cancel common factors to simplify further, but in this case, further simplification isn't possible. Simplifying expressions involves:

• Combining like terms when possible.
• Canceling common factors in the numerator and denominator.

Common Denominators

Identifying common denominators is crucial when doing operations with fractions. It allows you to add or subtract by ensuring each fraction refers to pieces of the same size. To find a common denominator, look for the least common multiple of each denominator in the fractions you’re adding or subtracting.
In our problem, all terms had the denominator \( y-5 \) already. When fractions share a denominator, you can directly add or subtract their numerators, as we did with the expression \( \frac{2y}{y-5} - \left( \frac{2}{y-5} + \frac{y-2}{y-5} \right) \). Having a common denominator simplifies your work dramatically:

• It prevents errors that can occur from inaccurate arithmetic with different denominators.
• Enables clearer comparisons and simplifications.

Ensuring common denominators keeps the calculations straightforward and understandable.