Problem 64 - Algebra 2

Question

Add or subtract. Simplify where possible. $$ \frac{5}{y+3}+\frac{15}{y-3} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{10(2y+3)}{y^2-9}\)

Step 1: Make the denominators of both fractions same

Multiply the numerator and denominator of the first term, \(\frac{5}{y+3}\), by \((y-3)\) and the second term, \(\frac{15}{y-3}\), by \((y+3)\) to make the denominators the same. The expression becomes \[\frac{5*(y-3)}{(y+3)*(y-3)} + \frac{15*(y+3)}{(y+3)*(y-3)}\] Simplify the expression to \[\frac{5y-15}{y^2-9} + \frac{15y+45}{y^2-9}\]

Step 2: Add the fractions

Since the denominators of both fractions are now the same (\(y^2-9\)), you can add the two fractions together: \[\frac{5y-15+15y+45}{y^2-9}\]This simplifies to \[\frac{20y+30}{y^2-9}\]

Step 3: Simplify the fraction

We can simplify the fraction by factorizing 10 out of the numerator, the expression becomes \[\frac{10(2y+3)}{y^2-9}\]

Key Concepts

Adding Rational Expressions

When adding rational expressions, it's essential to have a common denominator, just like with regular fractions. This makes it easier to combine them. In the given exercise, we have two rational expressions: \(\frac{5}{y+3}\) and \(\frac{15}{y-3}\). To add these together, we need the denominators to match.

Here's what you do:

• Identify the Denominators: Look at \((y+3)\) and \((y-3)\).
• Find the Common Denominator: Multiply each part of the expressions by the other expression's denominator, resulting in \((y+3)(y-3)\).
• Revise Each Expression: Multiply the top and bottom of each fraction by the necessary expression to get the common denominator.

Now you can add their numerators: \(5(y-3) + 15(y+3)\). Combining and expanding these terms lead to a common expression that can be easily added.

Simplifying Rational Expressions

Simplifying helps us make expressions easier to work with. Once the fractions are added, we end up with \(\frac{20y+30}{y^2-9}\). It's important to simplify this further to make it cleaner.

Here are the steps:

• Check the Numerator: The numerator \(20y+30\) can be factored to simplify the expression.
• Factor Out the Common Term: Notice that both terms share a common factor of 10, so factor that out: \(10(2y+3)\).

This leaves the expression as \(\frac{10(2y+3)}{y^2-9}\). Always check to see if things can be reduced further or if any common terms exist.

Factoring in Algebra

Factoring is a crucial skill in algebra, allowing us to break down expressions into simpler parts. In our exercise, we see factoring used several times to simplify the expression.

Here's how it works:

• Understanding Factoring: It's the process of writing an expression as a product of its factors.
• Applying to the Exercise: The expression \(y^2-9\) is a difference of squares, which can be factored as \((y+3)(y-3)\). This matches our denominators and helps simplify the process.
• Using Factoring to Reduce: In the expression \(20y+30\), we factor out a 10 to simplify: \(10(2y+3)\).

Factoring turns complex expressions into manageable parts, making solving much more straightforward. Always look for common factors or patterns, like differences of squares, for simplification.