Subtracting reciprocal expressions with x and y variables

Key takeaway: This problem checks whether you can subtract two reciprocals by using a common denominator and simplifying the numerator correctly.

Build the common denominator

We need to simplify

$\frac{1}{y} - \frac{1}{x}$.

The common denominator is $xy$, so rewrite each fraction:

$\frac{1}{y} = \frac{x}{xy}$

$\frac{1}{x} = \frac{y}{xy}$

Subtract the numerators

Now subtract:

$\frac{x}{xy} - \frac{y}{xy} = \frac{x-y}{xy}$.

So the simplest form is

$\boxed{\frac{x-y}{xy}}$.

Check the answer choices

This exact expression is not written correctly in the provided choices, but it is the correct algebraic simplification.

Important algebra note

Because subtraction is not commutative, $\frac{1}{y} - \frac{1}{x}$ is not the same as $\frac{1}{x} - \frac{1}{y}$. Switching the order would change the sign of the result.

Domain restriction

This expression is defined only when $x \neq 0$ and $y \neq 0$. Those restrictions come from the original denominators.

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Pitfalls the pros know 👇 A common mistake is to subtract the denominators directly and write something like $\frac{1-1}{y-x}$ or $\frac{1}{y-x}$. Fraction subtraction does not work that way. You must first rewrite both terms over a shared denominator. Another mistake is losing the sign and writing $\frac{y-x}{xy}$, which is the negative of the correct result. Order matters in subtraction, so keep the original sequence exactly as given.

What if the problem changes? If the expression were $\frac{1}{x} - \frac{1}{y}$ instead, the result would be $\frac{y-x}{xy}$. If it were $\frac{2}{y} - \frac{1}{x}$, the common denominator would still be $xy$, but the numerator would become $2x - y$, giving $\frac{2x-y}{xy}$. These variants all follow the same rule: rewrite both fractions over the same denominator before subtracting.

Tags: reciprocal expression, common denominator, fraction subtraction