Problem 58 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{x-y}{\frac{1}{y}-\frac{1}{x}} $$

Step-by-Step Solution

Answer

The simplified expression is \xy\.

Step 1: Simplify the Denominator

Observe that the denominator has two fractions. Combine the fractions in the denominator by finding a common denominator. The common denominator of \(\frac{1}{y}\) and \(\frac{1}{x}\) is \xy\.

Step 2: Combine Denominator Fractions

Rewrite \(\frac{1}{y} - \frac{1}{x}\) using the common denominator. This becomes: \(\frac{x - y}{xy}\).

Step 3: Simplify the Complex Fraction

We now have \(\frac{x-y}{\frac{x-y}{xy}}\). Simplify by multiplying the numerator by the reciprocal of the denominator: \(\frac{x-y}{1} \times \frac{xy}{x-y}\).

Step 4: Cancel Common Factors

The \(x-y\) terms cancel out, leaving \(xy\). So, the expression simplifies to \xy\.

Key Concepts

Common Denominator

When working with fractions, it's often necessary to combine or compare them, which requires a common denominator. The common denominator is the smallest number that both denominators can divide into without leaving a remainder. For example, in the fractions \(\frac{1}{y}\) and \(\frac{1}{x}\), the common denominator is \[xy\]. By rewriting each fraction with this common denominator, we can combine them into a single fraction. Here's how it works:

• \(\frac{1}{y}\) can be rewritten as \(\frac{x}{xy}\).
• \(\frac{1}{x}\) can be rewritten as \(\frac{y}{xy}\).

This allows us to subtract the fractions: \(\frac{x}{xy} - \frac{y}{xy} = \frac{x-y}{xy}\). Using a common denominator is critical for simplifying complex fractions.

Reciprocal

A reciprocal of a number is simply one divided by that number. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). Multiplying a number by its reciprocal always equals \(1\). In simplifying complex fractions, the reciprocal is used to clear fractions in the denominator. For instance, in the original problem, we had \(\frac{x-y}{\frac{x-y}{xy}}\). To remove the denominator \(\frac{x-y}{xy}\), we multiply by its reciprocal \(\frac{xy}{x-y}\). This effectively makes the fraction easier to handle:

• Original: \(\frac{x-y}{\frac{x-y}{xy}}\)
• Multiply by reciprocal: \(\frac{x-y}{1} \times \frac{xy}{x-y}\)

Utilizing reciprocals is essential for simplifying and solving complex fractions.

Cancel Common Factors

When simplifying fractions, canceling common factors—components that are alike in both the numerator and the denominator—is a helpful technique. If a term appears in both parts of a fraction, you can 'cancel' them out, effectively reducing the fraction. In our problem, after rewriting, we have \(\frac{x-y}{1} \times \frac{xy}{x-y}\). Notice the \(x-y\) term appears in both the numerator and denominator. This common factor can be canceled:

• \(\frac{x-y}{1} \times \frac{xy}{x-y} = xy\)

By canceling the \(x-y\) terms, the expression is simplified significantly. Always look for common factors to simplify fractions more efficiently.

Simplify Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form by eliminating any unnecessary terms or combining like terms. It often involves several steps, including using a common denominator, finding reciprocals, and canceling common factors. Here's a breakdown of simplifying the given problem:

• Start with: \(\frac{x-y}{\frac{1}{y}-\frac{1}{x}}\)
• Find a common denominator for the bottom fractions: \(\frac{x-y}{\frac{x-y}{xy}}\)
• Multiply the numerator by the reciprocal of the denominator: \(\frac{x-y}{1} \times \frac{xy}{x-y}\)
• Cancel out common factors \(x-y\): \(xy\)

Following these steps will consistently lead to a simplified algebraic expression, making it easier to understand and solve.