Problem 50 - College Algebra

Question

For the following exercises, simplify the rational expression. $$ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{x}{y}+\frac{y}{x}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(x-y)(x+y)}{x^2 + y^2} \).

Step 1: Identify the common denominator

The given expression is \( \frac{\frac{x}{y} - \frac{y}{x}}{\frac{x}{y} + \frac{y}{x}} \). Start by identifying the common denominator for both the numerator and denominator. The common denominator for both would be \( yx \).

Step 2: Rewrite the fractions over the common denominator

Rewrite the expression with \( yx \) as a common denominator:\[ \frac{\frac{x^2}{yx} - \frac{y^2}{yx}}{\frac{x^2}{yx} + \frac{y^2}{yx}} \]This simplifies to:\[ \frac{\frac{x^2 - y^2}{yx}}{\frac{x^2 + y^2}{yx}} \]

Step 3: Simplify the expression by cancelling

Since the fractions in both the numerator and denominator have \( yx \) as a common denominator, you can cancel \( yx \) from both the numerator and denominator. This gives:\[ \frac{x^2 - y^2}{x^2 + y^2} \]

Step 4: Factor the numerator, if possible

The numerator \( x^2 - y^2 \) can be factored using the difference of squares formula, \( a^2 - b^2 = (a-b)(a+b) \). Thus,\[ x^2 - y^2 = (x-y)(x+y) \]

Step 5: Final simplified expression

The simplified form of the rational expression is:\[ \frac{(x-y)(x+y)}{x^2 + y^2} \] There are no more common factors to cancel in the numerator and denominator.

Key Concepts

Common Denominator

When simplifying rational expressions, finding the common denominator is often a necessary step. A common denominator allows us to combine or compare fractions more easily.
For a given expression like \( \frac{\frac{x}{y} - \frac{y}{x}}{\frac{x}{y} + \frac{y}{x}} \), each fraction involves different denominators. The goal is to write each fraction over a single common denominator.

• First, determine the denominators involved. Here, they are \( y \) and \( x \).
• Next, the common denominator is simply the product of these individual denominators, which gives \( yx \).
• By rewriting both numerator and denominator expressions over \( yx \), they become comparable, allowing us to simplify the entire expression smoothly.

This key step is foundational in simplifying rational expressions and is especially useful when working with algebraic fractions.

Difference of Squares

The difference of squares is a powerful algebraic tool used to simplify certain expressions. The general formula is \( a^2 - b^2 = (a-b)(a+b) \).
This rule is highly effective when simplifying expressions where a difference of squares is present.
Let's look at the expression \( x^2 - y^2 \). In our given problem, after finding a common denominator and simplifying, our expression contains \( x^2 - y^2 \) in the numerator.
Using the difference of squares formula:

• Recognize \( x^2 - y^2 \) as a difference of squares.
• Factor it into \((x-y)(x+y)\).

Recognizing patterns like a difference of squares enables us to rewrite expressions in a more simplified or factorable form, leading to easier manipulation and eventual simplification.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This often includes finding common denominators, factoring, and canceling like terms.
When dealing with rational expressions like \( \frac{\frac{x}{y} - \frac{y}{x}}{\frac{x}{y} + \frac{y}{x}} \), simplification can initially seem complex.
However, it becomes manageable through a systematic approach:

• Find a common denominator, allowing us to rewrite fractions in a more compatible form.
• Cancel shared terms once they are written over the same denominator – in our case, the factor \( yx \) in the numerator and denominator gets canceled.
• Further simplify where possible by factoring expressions, such as applying the difference of squares rule where applicable.

By consistently applying these steps, any rational expression can be broken down into its simplest components, greatly easing the difficulty of solving or further manipulating algebraic equations.