Problem 54 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{q}{p-q}-\frac{q}{q-p}$$

Step-by-Step Solution

Answer

The result is \( \frac{2q}{p-q} \).

Step 1: Understand the Problem

We need to subtract two fractions: \( \frac{q}{p-q} \) and \( \frac{q}{q-p} \). The denominators are \( p-q \) and \( q-p \), which are negatives of each other.

Step 2: Recognize Equivalent Fractions

The term \( q-p \) is the negative of \( p-q \). Therefore, \( \frac{q}{q-p} = -\frac{q}{p-q} \). This means we can write the expression as \( \frac{q}{p-q} - \left(-\frac{q}{p-q}\right) \).

Step 3: Add the Fractions

Since the fractions now have the same denominator (\( p-q \)), we can combine them: \[ \frac{q}{p-q} + \frac{q}{p-q} = \frac{q+q}{p-q} = \frac{2q}{p-q}. \]

Step 4: Simplify the Result

The expression \( \frac{2q}{p-q} \) is already in its simplest form.

Key Concepts

Subtracting Fractions

Subtracting fractions involves finding a common denominator for the fractions in question. Once a common denominator is identified, the numerators can be directly subtracted while keeping the denominator the same. In the given problem, the fractions to subtract are \( \frac{q}{p-q} \) and \( \frac{q}{q-p} \). The takeaway here is understanding that fractions cannot be subtracted directly if their denominators differ. However, identifying that they are negatives of each other, as in this example, simplifies the process. By recognizing the relationship between the denominators, the problem of common denominators is inherently solved, allowing for an easier subtraction or addition of the numerators.

Equivalent Expressions

Creating equivalent expressions is a fundamental skill in algebra that allows us to manipulate and simplify expressions. In our exercise, recognizing that \( q-p = -(p-q) \) is key, as it creates equivalent expressions that can be easily combined. This recognition lets us rewrite the second fraction's denominator and adjust its sign: \( \frac{q}{q-p} = -\frac{q}{p-q} \). Hence, by transforming the subtraction of fractions into the addition of equivalent expressions, the problem becomes much more straightforward.

Recognizing equivalent expressions helps find common tasks, such as finding common denominators or simplifying fractions. By understanding equivalent forms, complex expressions often become simpler and more manageable.

Negative Denominators

Negative denominators can seem tricky, but they follow the same principles as positive denominators. Here we explore how they function in algebraic fractions. When comparing denominators like \( p-q \) and \( q-p \), we notice they are negatives of each other. In algebra, it is possible to multiply both the numerator and the denominator of a fraction by \(-1\), which will change the signs but leave the value unchanged.
This equivalency makes it possible to perform operations by flipping negative signs as needed.

By changing a negative denominator to its positive counterpart, or vice-versa, the fraction's value is preserved, all while standardizing the expression. This is a critical concept, allowing algebraic operations to proceed smoothly without changing the inherent value of the expressions involved.