Problem 67 - Elementary and Intermediate Algebra

Question

Perform the operations and simplify, if possible. See Example 7. $$\frac{3 y-2}{2 y+6}-\frac{2 y-5}{2 y+6}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{2}\).

Step 1: Identify the Denominator

Both fractions \( \frac{3y - 2}{2y + 6} \) and \( \frac{2y - 5}{2y + 6} \) have the same denominator \( 2y + 6 \). This means we can combine the fractions directly by operating on their numerators.

Step 2: Subtract the Numerators

Subtract the second numerator from the first: \((3y - 2) - (2y - 5)\). Ensure to distribute the negative sign across the terms in the second numerator resulting in: \(3y - 2 - 2y + 5\).

Step 3: Simplify the Numerator

Combine like terms from the expression \(3y - 2y + 5 - 2\). Simplify it to find \(y + 3\).

Step 4: Express the Resulting Simplified Fraction

Place the simplified numerator over the common denominator: \(\frac{y + 3}{2y + 6}\).

Step 5: Simplify the Fraction, if Possible

Factor any expressions, if possible. The denominator can be factored into \(2(y + 3)\). So the expression becomes \(\frac{y + 3}{2(y + 3)}\).

Step 6: Cancel Out Common Factors

Since \(y + 3\) is a common factor in both the numerator and the denominator, cancel it out. The fraction simplifies to \(\frac{1}{2}\).

Key Concepts

Numerator Operations

When handling algebraic fractions, especially when the denominators are the same, it becomes crucial to perform operations focusing on the numerators. For example, in the expression \[\frac{3y - 2}{2y + 6} - \frac{2y - 5}{2y + 6},\]we have a subtraction operation between the numerators of two fractions with a common denominator. The task here is to handle the numerators: \[(3y - 2) - (2y - 5).\]The negative sign in front of the second fraction needs careful distribution across the second numerator, changing it to \[3y - 2 - 2y + 5.\]To proceed, identify and combine like terms to simplify this expression. In this case, combine

• the terms with \(y\), which are \(3y - 2y\),
• the constant terms, which are \(-2 + 5\).

After simplification, you end up with \[y + 3.\]Understanding numerator operations helps streamline calculations when you have fractions sharing the same denominator.

Common Denominators

A common denominator is essential in combining two fractions. It ensures both fractions can be manipulated together efficiently. In our example, both fractions already possess the same denominator \[2y + 6.\]This commonality allows us to directly subtract the numerators, without needing further adjustments to the fractions themselves.When fractions have different denominators, you would have to find the least common denominator to combine them. However, since \[\frac{3y - 2}{2y + 6}\]and \[\frac{2y - 5}{2y + 6}\]share \(2y + 6\), we bypass this extra step.Leveraging common denominators simplifies processes like addition or subtraction of algebraic fractions, as it centralizes the focus to operations on the numerators alone. This lesson highlights the importance of identifying and using a common denominator to simplify the process and make problem-solving straightforward.

Simplifying Fractions

Simplifying fractions is crucial for yielding the most reduced form of an expression. After performing numerator operations, the expression \[\frac{y + 3}{2y + 6}\]requires further simplification.Firstly, consider factoring the denominator. In this example, \[2y + 6\]can be factored as \[2(y + 3).\]This factorization reveals that both the numerator, \[y + 3,\]and the denominator share a common factor: \[y + 3.\]Cancel out this common factor in the numerator and denominator, simplifying the fraction to \[\frac{1}{2}.\]The concept of simplifying fractions involves looking for factors in the numerator and denominator that can reduce the fraction to its simplest form. This skill ensures clearer, more concise expressions and often simplifies further algebraic equations or operations based on these expressions.