Problem 89 - Intermediate Algebra

Question

Simplify each expression. $$ \left(\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x+y}{x}}\right) \div\left(\frac{\frac{x^{2}}{y}-y}{\frac{y^{2}}{x}-x}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the First Fraction of the First Expression

Start by simplifying the numerator of the first expression \( \left(\frac{x}{y}-\frac{y}{x}\right) \). Rewrite the subtraction as a single fraction: \[ \frac{x^2 - y^2}{xy}. \]

Step 2: Simplify the Second Fraction of the First Expression

Now simplify the denominator of the first expression \( \left(\frac{x+y}{x}\right) \). The denominator can be expressed as: \[ \frac{x+y}{x}. \]

Step 3: Simplify the First Expression

Combine the simplified numerator and denominator from Steps 1 and 2 to get the first expression: \[ \frac{\frac{x^2 - y^2}{xy}}{\frac{x+y}{x}} = \frac{x^2-y^2}{xy} \times \frac{x}{x+y} = \frac{x(x^2-y^2)}{xy(x+y)}.\] Use the identity \( x^2 - y^2 = (x-y)(x+y) \) to rewrite: \[ \frac{(x-y)(x+y)}{xy(x+y)} = \frac{x-y}{y(x+y)}. \]

Step 4: Simplify the First Fraction of the Second Expression

Rewrite the numerator of the second expression \( \left(\frac{x^2}{y} - y \right) \) as a single fraction: \[ \frac{x^2 - y^2}{y}. \]

Step 5: Simplify the Second Fraction of the Second Expression

Now simplify the denominator of the second expression \( \left(\frac{y^2}{x} - x \right) \). The denominator can be expressed as: \[ \frac{y^2 - x^2}{x}. \]

Step 6: Simplify the Second Expression

Combine the simplified numerator and denominator from Steps 4 and 5 to get the second expression: \[ \frac{\frac{x^2 - y^2}{y}}{\frac{y^2 - x^2}{x}} = \frac{x^2-y^2}{y} \times \frac{x}{y^2-x^2} = -\frac{x^2-y^2}{y(y^2-x^2)}. \] Use the identity \( x^2 - y^2 = (x-y)(x+y) \): \[ - \frac{(x-y)(x+y)}{y(x-y)(x+y)} = -\frac{1}{y}. \]

Step 7: Divide the Simplified Expressions

Now divide the simplified first expression by the simplified second one: \[ \frac{\frac{x-y}{y(x+y)}}{-\frac{1}{y}} = \frac{x-y}{y(x+y)} \times -y = \frac{-(x-y)}{x+y}. \]

Key Concepts

Simplification

To simplify algebraic expressions means to reduce them to their most basic form, making complex expressions easier to work with or understand. The goal is to transform the expression without changing its value. In the exercise provided, simplification helps to neatly condense a large fraction into a more manageable form.

The process often involves:

• Combining like terms.
• Using algebraic identities to transform expressions.
• Cancelling common factors present in the numerator and the denominator.

In simplifying the first part of the expression, it was vital to recognize the difference of squares, such as in the identity \(x^2 - y^2 = (x-y)(x+y)\). This step is crucial as it allows the expression to be factored and reduced more effectively.

Additionally, ensuring each component, particularly in this exercise's fractions, is presented with a common denominator aids in simplifying the steps needed to solve or manage subsequent portions of the mathematical process.

Fractions

Understanding fractions is essential, as simplifying algebraic expressions often involves dealing with them directly. A fraction consists of a numerator and a denominator.

Key aspects involve:

• Recognizing common denominators to facilitate addition or subtraction.
• Multiplying or dividing fractions appropriately by flipping and multiplying (known as multiplying by the reciprocal).
• Simplifying complex fractions, which can occur when there are fractions within fractions.

In the exercise, fractions are used extensively. First, by recognizing certain expressions fit as fractions makes them easier to solve. For example, moving from \( \frac{x}{y} - \frac{y}{x} \) to \( \frac{x^2-y^2}{xy} \) is a classic fraction simplification approach using common denominators, which helps not only with subtraction but also simplifies further operations in the expression. Similarly, understanding how to adjust numerators and denominators ensures fractions are correctly transformed during simplification.

Rational Expressions

Rational expressions are algebraic fractions where both the numerator and the denominator are polynomials. They follow the same principles as regular fractions but often involve more intricate polynomials.

In the given exercise, dividing one rational expression by another is central to the solution.

It's important to:

• Simplify each piece fully before performing division.
• Identify common factors to facilitate cancellation, making the expression simpler.
• Apply rational expression operations correctly, such as multiplying by the reciprocal.

For instance, the initial complex structure divides the first rational expression by a second, which involves carefully flipping the second expression and applying it as a multiplication. This step must be executed after fully simplifying both the numerator and the denominator independently. Such processes showcase how they can be managed to yield a less complicated but equivalent expression, demonstrating the power of rational expression manipulation in algebra.