Perform the indicated operation, and write each answer in lowest terms. (4/x-y) - (9/x-y)

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Identify the given expression: \(\frac{4}{x - y} - \frac{9}{x - y}\).

Since both fractions have the same denominator \((x - y)\), you can combine the numerators directly over the common denominator: \(\frac{4 - 9}{x - y}\).

Simplify the numerator by performing the subtraction: \(4 - 9 = -5\), so the expression becomes \(\frac{-5}{x - y}\).

Rewrite the expression to make it clearer, for example as \(-\frac{5}{x - y}\).

Check if the fraction can be simplified further by factoring numerator or denominator, but since \(-5\) is a constant and \(x - y\) is already simplified, this is the expression in lowest terms.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Like Terms and Common Denominators

When subtracting rational expressions, it is essential to identify if the denominators are the same. If the denominators are identical, the numerators can be directly subtracted while keeping the denominator unchanged. This simplifies the operation and avoids unnecessary complexity.

Subtracting Rational Expressions

Subtracting rational expressions involves subtracting the numerators and keeping the denominator the same, provided the denominators are equal. If denominators differ, you must find a common denominator before performing the subtraction. This ensures the expressions are combined correctly.

Simplifying Rational Expressions

After performing the subtraction, simplify the resulting expression by factoring numerators and denominators and canceling common factors. Writing the answer in lowest terms means reducing the expression to its simplest form, which makes it easier to interpret and use in further calculations.

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