Problem 42 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{5 y}{6}-\frac{3 y}{8}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{11y}{24} \).

Step 1: Identify the denominators

The denominators of the fractions are 6 and 8. To add or subtract fractions, they must have a common denominator.

Step 2: Find the lowest common denominator (LCD)

The lowest common multiple of 6 and 8 is 24. This will be the common denominator.

Step 3: Adjust the fractions to have the same denominator

For \( \frac{5y}{6} \), multiply both the numerator and the denominator by 4: \( \frac{5y \times 4}{6 \times 4} = \frac{20y}{24} \).For \( \frac{3y}{8} \), multiply both the numerator and the denominator by 3: \( \frac{3y \times 3}{8 \times 3} = \frac{9y}{24} \).

Step 4: Subtract the fractions

Now that both fractions have a common denominator, subtract the numerators: \( \frac{20y}{24} - \frac{9y}{24} = \frac{(20y - 9y)}{24} = \frac{11y}{24} \).

Step 5: Simplify the fraction

Check if \( \frac{11y}{24} \) can be simplified. As 11 is a prime number and does not divide evenly into 24, \( \frac{11y}{24} \) is already in simplest form.

Key Concepts

Lowest Common Denominator

When working with fractions, especially in algebra, it's crucial to have a common denominator before performing operations like addition or subtraction. This common ground allows us to directly combine the numerators. The "lowest common denominator" (LCD) is the smallest multiple that both denominators share.

• To find the LCD, begin by listing the multiples of each denominator.
• Select the smallest number that appears in both lists as your LCD.

In our example, with denominators 6 and 8, the smallest common multiple is 24. By converting each fraction to share this denominator, you enable seamless operations between them.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and denominator have no common factors other than 1. This process makes fractions easier to understand and work with.

• First, identify if the numerator and the denominator have any common factors.
• Divide both the numerator and the denominator by their greatest common factor (GCF).
• If no simplification is possible, then the fraction is already in its simplest form.

In our solved exercise, the result was \( \frac{11y}{24} \). Since 11 is a prime number and does not share any factors with 24, the fraction is already simplified to its simplest form.

Algebraic Expressions with Fractions

Fractions in algebra function similarly to those in arithmetic but include variables in their numerators or denominators. Working with these expressions requires careful manipulation to maintain balance in equations.

• Determine if variables appear in the numerator, denominator, or both.
• Follow the same rules for finding LCDs and simplifying as you would with standard fractions.
• Perform operations, like addition or subtraction, with precision to avoid errors with variables involved.

For an expression like \( \frac{5y}{6} - \frac{3y}{8} \), the variable \( y \) moves along with its respective coefficients during each operation. By adjusting each fraction to have the LCD, you treat the variables similarly to constants, ensuring the final result accurately represents the entire expression.