Problem 23 - Intermediate Algebra

Question

Simplify each complex fraction. See Example 4. $$ \frac{\frac{y}{x}-\frac{x}{y}}{\frac{1}{x}+\frac{1}{y}} $$

Step-by-Step Solution

Answer

The complex fraction simplifies to \( y - x \).

Step 1: Simplify the Numerator

The numerator of the complex fraction is \( \frac{y}{x} - \frac{x}{y} \). To combine these fractions, find a common denominator. The common denominator for \( x \) and \( y \) is \( xy \). Rewrite each fraction with this common denominator: \( \frac{y^2}{xy} - \frac{x^2}{xy} = \frac{y^2 - x^2}{xy} \).

Step 2: Simplify the Denominator

The denominator of the complex fraction is \( \frac{1}{x} + \frac{1}{y} \). Similarly, find a common denominator, which is \( xy \). Rewrite the sum: \( \frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy} \).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now, divide the expression from Step 1 by the expression from Step 2: \( \frac{\frac{y^2 - x^2}{xy}}{\frac{y + x}{xy}} \). Division by a fraction is the same as multiplying by its reciprocal, so this becomes \( \frac{y^2 - x^2}{xy} \times \frac{xy}{y + x} \).

Step 4: Simplify the Expression

Cancel out \( xy \) in the fraction since it appears in both the numerator and the denominator: \( \frac{y^2 - x^2}{y + x} \). Recognize that the numerator \( y^2 - x^2 \) is a difference of squares and can be factored: \( (y + x)(y - x) \). Thus, the expression simplifies to \( \frac{(y + x)(y - x)}{y + x} \). Further simplification gives \( y - x \) after canceling \( y + x \) from the numerator and denominator.

Key Concepts

Simplifying Fractions

Simplifying fractions is a technique used to make fractions easier to understand or work with. A fraction consists of a numerator, the top number, and a denominator, the bottom number.
There are two main steps in simplifying fractions:

• Find a common factor: Calculate the greatest common factor (GCF) of the numerator and the denominator and divide both by this number.
• Reduce the fraction: The fraction is simplified when there are no common factors between the numerator and the denominator other than 1.

In the case of complex fractions, as in the given exercise, the numerator or denominator themselves are fractions.
Here's the key to simplifying them:

• Find a common denominator for any fractions in the numerator and the denominator separately.
• Once the fractions have a common denominator, add or subtract them to form a single simple fraction.
• The complex fraction can then be simplified by dividing or multiplying as needed.

This step-by-step approach makes handling complex fractions considerably less intimidating and more manageable.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operational symbols. Variables are essential, as they represent unknown values that we can manipulate within the expression.
When working with algebraic expressions, understanding terms is crucial:

• Terms in an expression: Terms are the individual components separated by addition or subtraction signs. Each term can include variables, coefficients (numbers in front of variables), and constants (independent numbers).
• Combining like terms: Like terms have the same variables and exponents. They can be combined by adding or subtracting their coefficients.
• Simplifying algebraic expressions: Factor or rearrange terms to make the expression easier to work with.

In our case, when dealing with the given complex fraction, we worked with the algebraic expressions like \(\frac{y}{x} - \frac{x}{y}\) and \(\frac{1}{x} + \frac{1}{y}\).
Understanding how these expressions interact allows us to simplify them effectively into a single line of expression.

Factoring Differences of Squares

The concept of factoring differences of squares is a fundamental idea in algebra that applies when simplifying expressions. A difference of squares is an expression formed like \(a^2 - b^2\).
This difference can be factored using a specific formula:

• The formula is \(a^2 - b^2 = (a + b)(a - b)\).
• You use this pattern to split the square terms into two binomial factors.

This concept was applied in our exercise when simplifying the expression \(y^2 - x^2\) into \((y + x)(y - x)\).
By factoring the expression, you can often simplify further by canceling terms that appear in both the numerator and the denominator.
Understanding this essential factorization technique helps to dismantle complex algebraic structures, simplifying them into manageable parts.