Problem 51 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{6 y}{y+4}+\frac{2 y}{y+3} $$

Step-by-Step Solution

Answer

Answer: The sum of the rational expressions is \(\frac{8y^2 + 26y}{(y+4)(y+3)}\).

Step 1: Find the Least Common Denominator (LCD)

First, we need to find the least common denominator (LCD) of the two expressions. The denominators of the given fractions are \((y+4)\) and \((y+3)\). Since these expressions have no common factors other than 1, their LCD will be the product of the two denominators: \((y+4)(y+3)\).

Step 2: Rewrite the fractions with the same denominator

To rewrite each fraction with the common denominator, we need to multiply the numerator and denominator of each fraction by the appropriate factors so that their denominators become equal to the LCD. For the first fraction: \(\frac{6y}{y+4}\), multiply the numerator and denominator by \((y+3)\): $$ \frac{6y}{y+4} \cdot \frac{y+3}{y+3} = \frac{6y(y+3)}{(y+4)(y+3)} $$ For the second fraction: \(\frac{2y}{y+3}\), multiply the numerator and denominator by \((y+4)\): $$ \frac{2y}{y+3} \cdot \frac{y+4}{y+4} = \frac{2y(y+4)}{(y+3)(y+4)} $$ Now the fractions have the same denominator: $$ \frac{6y(y+3)}{(y+4)(y+3)} + \frac{2y(y+4)}{(y+3)(y+4)} $$

Step 3: Add the fractions

Since the two fractions have the same denominator, we can simply add their numerators and keep the denominator the same: $$ \frac{6y(y+3) + 2y(y+4)}{(y+4)(y+3)} $$

Step 4: Simplify the numerator

Expand and simplify the numerator by distributing the factors: $$ \frac{6y^2 + 18y + 2y^2 + 8y}{(y+4)(y+3)} $$ Combine like terms: $$ \frac{8y^2 + 26y}{(y+4)(y+3)} $$

Step 5: Final answer

The simplified sum of the given rational expressions is: $$ \frac{8y^2 + 26y}{(y+4)(y+3)} $$

Key Concepts

Least Common Denominator

When working with fractions, especially rational expressions, it's crucial to have a common denominator for addition or subtraction. The least common denominator (LCD) is the smallest expression that all denominators in the problem can divide into evenly. In this exercise, the rational expressions have denominators of \( (y+4) \) and \( (y+3) \).

These two denominators have no common factors other than 1. Therefore, their LCD is simply the product of the two denominators: \((y+4)(y+3)\).

To find the LCD:

• Identify the denominators.
• Determine if there are any common factors.
• If there aren't, multiply the denominators together to get the LCD.

Finding the LCD allows us to rewrite the fractions so they have the same denominator, which simplifies the process of adding or subtracting them.

Adding Fractions

Once the fractions have the same denominator, we can easily add them together. In this scenario, each rational expression has been rewritten to share the common denominator \((y+4)(y+3)\). This step ensures both fractions can be combined directly by simply adding their numerators.

Here are the steps to add fractions with a common denominator:

• Rewrite each fraction so they have the same denominator if necessary (using multiplication).
• Add the numerators together and keep the common denominator.
• Simplify the result if possible.

For the provided exercise, the numerators are combined as follows: \[ \frac{6y(y+3) + 2y(y+4)}{(y+4)(y+3)} \] You take \( 6y(y+3) \) and \( 2y(y+4) \) and add them together while maintaining the shared denominator.

Simplifying Expressions

After adding the fractions, the resulting expression often requires simplification. Simplifying makes the expression easier to understand and use in further calculations. This process usually involves expanding the terms, combining like terms, and reducing the expression if possible.

In our example, the expanded numerator becomes: \[ 6y^2 + 18y + 2y^2 + 8y \]
The next step is to combine like terms:

• Combine terms with the same variable and exponent.
• Rearrange the terms in descending order of exponent.

These steps lead us to the simplified form: \[ \frac{8y^2 + 26y}{(y+4)(y+3)} \] Ensure there are no common factors in the numerator and denominator that can be further reduced. This process helps clarify the result and is essential in reaching the simplest form of any rational expression.