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Chapter 6: Problem 69

Add or subtract as indicated. $$\frac{w}{w-z}-\frac{z}{z-w}$$

Short Answer

Expert verified
\( \frac{w + z}{w - z} \)

Step by step solution

01

Identify the common denominator

Rewrite each fraction so they have a common denominator. Notice that \(\frac{w}{w-z}\) and \(\frac{z}{z-w}\) have similar denominators but one is the negative of the other.
02

Adjust the sign of one denominator

Since \(z-w = -(w-z)\), change the denominator of the second fraction by multiplying the numerator and the denominator by \(-1\): \(\frac{z}{z-w} = \frac{z \times -1}{(z-w) \times -1} = \frac{-z}{w-z}\). Now the expression is \(\frac{w}{w-z} - \frac{-z}{w-z}\).
03

Combine the fractions

Now that both fractions have the same denominator, combine them: \(\frac{w - (-z)}{w-z}\).
04

Simplify the numerator

Simplify the numerator \(w - (-z) = w + z\). Therefore, the expression becomes \(\frac{w + z}{w-z}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common denominator
Understanding the concept of a common denominator is crucial when working with rational expressions and fractions. A common denominator is a shared multiple of the denominators of two or more fractions. It allows us to combine the fractions easily. In the case of \(\frac{w}{w-z} \) and \(\frac{z}{z-w}\), we observe that their denominators \(w-z\) and \(z-w\) are negatives of each other. To find a common denominator, we can use the relationship \(z-w = -(w-z)\). By rewriting \(\frac{z}{z-w}\) to \(\frac{-z}{w-z}\), we have matched the denominators, making them the same.
combining fractions
Once we have a common denominator, we can combine the fractions by adding or subtracting the numerators while keeping the denominator the same. For the expression \(\frac{w}{w-z} - \frac{z}{z-w}\), adjusting the second fraction gives us \(\frac{w}{w-z} - \frac{-z}{w-z}\). Combining these fractions involves performing subtraction on the numerator: \( w - (-z) \). When we combine, we essentially get: \(\frac{w - (-z)}{w-z}\).
simplifying expressions
Simplifying expressions is about making them as simple as possible. This often involves combining like terms or reducing fractions. In the context of our example, after combining the fractions, we had \(\frac{w - (-z)}{w-z}\). Subtracting a negative is the same as adding, so \(w - (-z) = w + z\). This simplifies our fraction to: \(\frac{w+z}{w-z}\). The numerator \(w+z\) and the denominator \(w-z\) can't be further simplified, so this is our final simplified expression.

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