Question

If \(x y \neq 0, \frac{1-x}{x y}=\) a. \(\frac{1}{x y}-\frac{1}{y}\) b. \(\frac{x}{y}-\frac{1}{x}\) c. \(\frac{1}{x y}-1\) d. \(\frac{1}{x y}-\frac{x^{2}}{y}\) e. \(\frac{1}{x}-\frac{1}{y}\)

Short answer

Option a: \( \frac{1}{xy} - \frac{1}{y} \)

Step-by-step solution

Understand the Problem

The exercise asks to simplify the expression \( \frac{1-x}{xy} \) and match it with one of the given choices: a. \( \frac{1}{xy} - \frac{1}{y} \), b. \( \frac{x}{y} - \frac{1}{x} \), c. \( \frac{1}{xy} - 1 \), d. \( \frac{1}{xy} - \frac{x^2}{y} \), e. \( \frac{1}{x} - \frac{1}{y} \).

Rewrite the Numerator

Rewrite the numerator of the fraction: \( 1 - x \). This does not need any changes.

Distribute the Denominator

Distribute the denominator \( xy \) over the numerator: \( \frac{1}{xy} - \frac{x}{xy} \). This splits the fraction into two separate fractions.

Simplify Each Fraction

Simplify each fraction: \( \frac{1}{xy} - \frac{x}{xy} = \frac{1}{xy} - \frac{1}{y} \). This shows the more simplified form.

Match with Options

Compare the simplified form \( \frac{1}{xy} - \frac{1}{y} \) with the given options. It matches option a. \( \frac{1}{xy} - \frac{1}{y} \).

Key concepts

Algebraic Simplification

Algebraic simplification involves rewriting expressions to make them easier to work with or understand. The key is to maintain the equality of the expression while making it more straightforward. This is often done by distributing, combining like terms, or canceling common factors.

In this exercise, we simplified the expression \(\frac{1-x}{xy}\). The primary steps involved:

• Identifying the numerator and denominator correctly.
• Distributing the denominator over the numerator.
• Simplifying each term individually.

Starting with \(\frac{1-x}{xy}\), this can be split into \(\frac{1}{xy} - \frac{x}{xy}\). Rewriting it this way helps to see that further simplification leads to \(\frac{1}{xy} - \frac{1}{y}\), which is option a. Effectively breaking down expressions is a powerful algebraic skill.

Remember, careful distribution and term-by-term simplification can reveal the cleanest form of an algebraic expression.

Fractions

Fractions represent parts of a whole and are crucial in many areas of math, including algebra. Simplifying fractions means breaking them down into their most basic components.

When handling fractions in algebra, follow these main strategies:

• Identify numerators and denominators.
• Look for opportunities to split fractions.
• Simplify each term separately.

In our problem, we started with the fraction \(\frac{1-x}{xy}\). By splitting it, we got \(\frac{1}{xy}\) and \(\frac{x}{xy}\).

Simplifying fractions in standard algebra often involves:

• Making the denominator common.
• Canceling out terms where applicable.

This way, \(\frac{1}{xy} - \frac{x}{xy}\) simplifies to \(\frac{1}{xy} - \frac{1}{y} = \frac{1}{xy} - \frac{1}{y}\). Mastering fraction simplification makes complex algebraic problems much easier to solve.

Standardized Test Prep

Success in standardized tests like the GRE relies heavily on efficient problem-solving skills. This includes effective math strategies such as algebraic simplification and managing fractions.

Here are some tips to excel at these topics:

• Practice breaking down complex expressions into simpler forms.
• Speed up skills by practicing fraction simplification.
• Always compare your simplified answer with given options to find the closest match.

In our example, efficiently simplifying \(\frac{1-x}{xy}\) and matching it with the correct option is crucial for saving time. It showed up as \(\frac{1}{xy} - \frac{1}{y}\), which matched option a.

Regular practice of GRE math problems—specifically focusing on speed and accuracy in algebra and fractions—can significantly improve test performance. Understanding fundamental concepts reduces error rates and enhances confidence, which is key for test-day success.