Problem 59 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x-y}+\frac{y}{y-x}$$

Step-by-Step Solution

Answer

The simplified result of the given expression is 1.

Step 1: Rewrite the Second Fraction.

Observe that \(y-x\) is the same as \(-(x-y)\). This means the denominator of the second fraction can be rewritten as a negative version of the first fraction's denominator. So we get the expression: \(\frac{x}{x-y} - \frac{y}{x-y}\)

Step 2: Sum the Fractions.

Now that the denominators of both fractions are the same, they can be added together. The expression can now be written as: \(\frac{x-y}{x-y}\)

Step 3: Simplification

As a rule of mathematics, any nonzero number divided by itself equals 1. Therefore, \(\frac{x-y}{x-y}\) simplifies to 1

Key Concepts

Additive Inverses

In algebra, the concept of additive inverses is quite important when dealing with fractions. Additive inverses are pairs of numbers that, when added together, equal zero. This is crucial when working with fractions since opposite denominators can be recognized using this property. For example, in the exercise, we have the denominators \(x-y\) and \(y-x\). These are additive inverses because \((x-y)\) is the negative of \((-1)\) times \((y-x)\). To visualize:

• The additive inverse of \(x-y\) is \(-1 \times (x-y)\) which simplifies to \(-x+y\) or \(y-x\).
• Using this inverse allows us to manipulate and rewrite expressions to enable easier calculations.

Recognizing and utilizing additive inverses can greatly simplify the process of adding and subtracting algebraic fractions. By making the denominators in our fractions uniform through this method, we can streamline our computations and simplify the fractions effectively.

Simplifying Fractions

Simplifying fractions is an important step in any problem involving algebraic fractions. We simplify fractions to reduce them to their simplest form, which makes them easier to understand and work with. In the provided exercise, once we achieved common denominators, we found ourselves with the fraction \(\frac{x-y}{x-y}\). Here’s how simplification works:

• The main principle in simplifying fractions is that any expression in the form \(\frac{a}{a}\), where \(a\) is non-zero, equals 1.


• In this case, \(x-y\) is the numerator and the denominator, which means we can reduce \(\frac{x-y}{x-y}\) straightforwardly to 1.


• This step is crucial in ensuring that the final answer is represented in the simplest form possible, removing any unnecessary complexity in the solution.

By applying these principles, we not only solve the problem at hand but also improve our ability to handle more complex algebraic fractions in the future.

Denominator Manipulation

Denominator manipulation is a key skill in algebra especially when dealing with the addition or subtraction of fractions that do not initially share the same denominator. In the original exercise, denominator manipulation was pivotal. We began with two fractions: \(\frac{x}{x-y}\) and \(\frac{y}{y-x}\). Here are the key steps we followed:

• Notice that \(y-x\) can be thought of as \(-(x-y)\) which allows the redirection of our equation.


• By rewriting the fraction \(\frac{y}{y-x}\) as \(-\frac{y}{x-y}\), both fractions now share the \(x-y\) denominator.


• This makes it possible to add or subtract these fractions directly, thus simplifying later steps in our solution.

The manipulation of denominators is a fundamental step in simplifying algebraic expressions. It leverages the properties of numbers and ensures that calculations are as straightforward and consistent as possible, allowing us to simplify even complex expressions systematically.