Problem 44 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{y+6}{5}-\frac{y-6}{5}$$

Step-by-Step Solution

Answer

\( \frac{12}{5} \)

Step 1: Subtract Fractions with Common Denominator

When subtracting fractions that have a common denominator, you can subtract directly across the numerators. The denominator remains the same. Here, the fractions are \( \frac{y+6}{5} \) and \( \frac{y-6}{5} \). Therefore, we have: \[ \frac{(y+6) - (y-6)}{5} \]

Step 2: Simplify the Numerator

Now simplify the expression in the numerator. Distribute the negative sign through the second term: \[ (y + 6) - (y - 6) = y + 6 - y + 6 \] Combine like terms: \[ y - y + 6 + 6 = 0 + 12 = 12 \] The numerator simplifies to 12.

Step 3: Write the Simplified Fraction

With the numerator simplified, write the fraction using the unchanged denominator: \[ \frac{12}{5} \]

Key Concepts

Common Denominator

A common denominator is crucial when working with fractions, especially during addition or subtraction. Having the same denominator allows you to directly perform operations on the numerators while keeping the denominator constant.
For instance, in the given problem \( \frac{y+6}{5} - \frac{y-6}{5} \), both fractions share the same denominator of 5. Hence, subtraction becomes straightforward as we only need to deal with the numerators.
Just like maintaining the same unit makes comparing numbers easy, a common denominator simplifies calculations, ensuring accurate results.

• To find a common denominator, ensure that both denominators are identical before performing addition or subtraction.
• If the fractions have different denominators, you may need to find the least common denominator (LCD).
• This exercise highlights a simpler case, where the denominators are already equal, simplifying the fraction operation to just subtracting the numerators.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, often making computations easier. Applying operations to numerators, while holding the denominator constant, is a critical skill.
In our example, we simplify \( (y+6) - (y-6) \)
by distributing the negative sign across the second bracket. We then combine like terms:
- Subtract or add the coefficients of similar terms, like terms "y" and constants.- This approach reduces complex expressions, making them more manageable.
After working through the exercise, the expression simplifies to \( \frac{12}{5} \).

• Always seek opportunities to combine like terms when dealing with algebraic fractions.
• Simplifying fractions improves clarity, making further mathematical operations with these fractions much easier.
• Ensure each step logically follows, maintaining accuracy as you work through the algebraic simplification process.

Algebraic Expressions

Algebraic expressions combine variables, numbers, and operations, serving as a foundational aspect in algebra. Understanding how to manage and manipulate these expressions is crucial in diverse mathematical tasks.
With the exercise example \( \frac{y+6}{5} - \frac{y-6}{5} \), the part \( y + 6 \) and \( y - 6 \) signify algebraic expressions. Simplifying these requires understanding the principles of algebra.

• Variables represent unknowns that can change, while constants remain fixed.
• Operations such as addition or subtraction are routine in manipulating these expressions.
• Effectively combining like terms simplifies these expressions, as demonstrated when subtracting or adding terms while keeping variables intact.

Mastering algebraic expressions enhances your ability to solve equations and understand diverse mathematical concepts, laying the groundwork for more advanced study.