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Chapter 7: Problem 13

Add or subtract. Write answer in lowest terms. \(\frac{x}{x+y}+\frac{y}{x+y}\)

Short Answer

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Step by step solution

01

- Identify common denominators

Notice that both fractions \(\frac{x}{x+y}\) and \(\frac{y}{x+y}\) have the same denominator \((x + y)\).
02

- Combine the numerators

Since the denominators are identical, add the numerators: \(\frac{x}{x+y} + \frac{y}{x+y} = \frac{x + y}{x+y}\).
03

- Simplify the fraction

Divide the numerator by the denominator: \(\frac{x + y}{x + y} = 1\).

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Key Concepts

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

Common Denominators
When adding or subtracting fractions, the first step is to identify if the fractions share a common denominator. The denominator is the bottom part of the fraction. If the denominators are the same, you can combine the fractions easily. In the exercise given, the fractions are \(\frac{x}{x+y}\) and \(\frac{y}{x+y}\). Both fractions have the same denominator, which is \(x+y\). This allows us to combine the numerators directly in the next step. Finding common denominators is crucial because it ensures we are adding or subtracting parts of the same whole. If the denominators were different, we would need to find a common denominator before combining the fractions.
Adding Fractions
Once the common denominator is identified, the next step is to add the fractions. We do this by adding their numerators while keeping the common denominator the same. In the provided exercise, after recognizing that both denominators are \(x+y\), we add the numerators: \(x\) and \(y\). This gives us:
  • \(\frac{x}{x+y} + \frac{y}{x+y} = \frac{x+y}{x+y}\)
By combining the numerators over the common denominator, the expression becomes simpler and ready for the final step of simplification. Adding fractions is straightforward once the common denominators are found. Always remember to keep the denominator the same while you only add or subtract the numerators.
Simplifying Fractions
The final step is simplifying the fraction. Simplifying means rewriting the fraction in its lowest terms. In our case, the fraction after adding the numerators is \(\frac{x+y}{x+y}\). Since the numerator and the denominator are the same, they cancel each other out, simplifying the fraction to 1:
  • \(\frac{x+y}{x+y} = 1\)
Simplifying fractions often involves dividing the numerator and the denominator by their greatest common divisor. However, in this example, because the numerator and denominator are identical, they simplify directly to 1. Always check if the numerator and denominator can be reduced further by a common factor for final simplification. This helps in achieving the fraction's simplest form.

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