Problem 78 - Introductory Algebra for College Students

Question

Explain how to subtract rational expressions when denominators are the same. Give an example with your explanation.

Step-by-Step Solution

Answer

To subtract two rational expressions with the same denominator, keep the denominator and simply subtract the numerators. For instance, \(\frac{5}{7} - \frac{3}{7} = \frac{2}{7}\).

Step 1: Identify Like Denominators

The first step is to identify if the rational expressions have the same denominators, this is important as it allows us to carry out the subtraction operation. Suppose we have two rational expressions \(\frac{a}{d}\) and \(\frac{b}{d}\) then both expressions have the same denominator (\(d\)).

Step 2: Subtract the Numerators

Given that the denominators are the same, we keep the denominator and subtract the numerators of the two expressions. In this case, we subtract \(b\) from \(a\). Thus the expression becomes \(\frac{a - b}{d}\).

Step 3: Simplify the Resulting Expression

Simplify the resulting rational expression by reducing it to its lowest terms. This often involves factoring the numerator and denominator, and then cancelling out common factors.

Step 4: Example

Let's consider an example: \(\frac{5}{7} - \(\frac{3}{7}\). Here, \(\frac{5}{7}\) and \(\frac{3}{7}\) have the same denominator, so we subtract the numerators: \(\frac{5 - 3}{7} = \(\frac{2}{7}\), which is the answer and doesn't need further simplification.