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

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

Step-by-step solution

Understand the question

The goal is to simplify the expression \(\frac{1-x}{xy}\) and find which option it matches in the given choices.

Rewrite the expression

Rewrite the given expression \(\frac{1-x}{xy}\) by splitting the numerator: \(\frac{1 - x}{xy} = \frac{1}{xy} - \frac{x}{xy}\).

Simplify the fractions

Simplify each term: \(\frac{1}{xy}\) remains as is, and \(\frac{x}{xy} = \frac{1}{y}\).

Combine the simplified fractions

Combine the simplified fractions from Step 3: \(\frac{1}{xy} - \frac{1}{y}\).

Match with the given options

Compare the result \(\frac{1}{xy} - \frac{1}{y}\) with the given options. The expression matches option (a).

Key concepts

fraction simplification

In algebra, simplifying fractions is crucial for problem solving. Consider the exercise's fraction \(\frac{1-x}{xy}\). Simplifying fractions means rewriting them in their simplest form. In this case, we can split the numerator to make it easier. This results in \(\frac{1 - x}{xy} = \frac{1}{xy} - \frac{x}{xy}\). Each term now can be simplified individually.

algebraic expressions

Algebraic expressions involve variables and constants, combined using mathematical operations. Here, the expression is \(\frac{1-x}{xy}\). Important steps when dealing with such an expression include:

• Understanding the expression: Identify components and their relationships.
• Applying operations: Use algebraic rules to manipulate the expression.
• Simplifying the result: Make the expression easier to handle.

In our problem, we applied these steps to break down and simplify the fraction.

mathematical problem solving

Solving mathematical problems often requires a step-by-step approach. The problem at hand involves:

• Identifying the goal: Simplifying \(\frac{1-x}{xy}\).
• Breaking down the problem: Splitting the numerator as \(\frac{1}{xy} - \frac{x}{xy}\).
• Simplifying individual parts: \(\frac{1}{xy}\text{ remains, and }\frac{x}{xy} = \frac{1}{y}\).
• Recombining and comparing: Combine terms to get \(\frac{1}{xy} - \frac{1}{y}\) and match it to the provided options.

By following these steps systematically, we simplify complex expressions and find correct solutions.

standardized test preparation

Preparing for standardized tests like the GRE involves practicing problems similar to our exercise. Here are some tips:

• Understand instructions: Read questions carefully and know what is asked.
• Practice similar problems: Work on exercises that target your weak areas.
• Simplify systematically: Use breakdown techniques to simplify expressions step by step.
• Check your work: Verify that your simplified expression matches one of the given options.

This type of structured practice helps you tackle different question types efficiently and improves your problem-solving skills.