### Adding Rational Expressions with Opposite Denominators#### Example Problem:Add: $ \frac{x}{x-y} + \frac{y}{y-x} $Since the denominators are opposites, either one can serve as the Least Common Denominator (LCD). If we choose $ x - y $, we can multiply $ \frac{y}{y - x} $ by $ \frac{-1}{-1} $ to build it into an equivalent rational expression with the denominator $ x - y $.#### Step-by-Step Solution:1. **Initial Expression**: \[ \frac{x}{x-y} + \frac{y}{y-x} \] 2. **Equating Denominators**: Since the denominators are opposites, \[ \frac{x}{x-y} + \frac{y}{y-x} = \frac{x}{x-y} + \frac{y}{y-x} \cdot \frac{-1}{-1} \] 3. **Multiplication**: \[ \frac{x}{x-y} + \frac{y \cdot (-1)}{(y-x) \cdot (-1)} \] 4. **Rewrite Denominator**: Rewrite $ -y + x $ as $ x - y $: \[ \frac{x}{x-y} + \frac{-y}{x-y} \] 5. **Combining the Numerators**: Add the numerators while keeping the common denominator: \[ \frac{x + (-y)}{x - y} \] 6. **Simplify**: \[ \frac{x - y}{x - y} = 1 \]The final simplified result is $ 1 $.#### Another Problem to Solve:Add: $ \frac{5a}{a-b} + \frac{b}{b-a} $Complete this problem using the same steps outlined above. Don’t forget to rewrite the denominators to a common term before combining the numerators and simplifying.

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Description: ### Adding Rational Expressions with Opposite Denominators#### Example Problem:Add: $ \frac{x}{x-y} + \frac{y}{y-x} $Since the denominators are opposites, either one can serve as the Least Common Denominator (LCD). If we choose $ x - y $, we can

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Add: Since the denominators are opposites, either one can serve as the LCD. If we choose x – y, we can multiply Y. by = to build it into an equivalent rational expression with the denominator x - y. When y - x is multiplied by -1, the subtraction is reversed, and the result is x - y. +Y y - x so that it has a denominator of x - y. + Build x - y Multiply the numerators. Multiply the denominators. %3D + X - y Rewrite the second denominator, -y + x, as x - y. The fractions now have a common denominator. x - y x - y Add the numerators. Write the difference over the common denominator x - y. x - y Simplify. 5a Add: a b – a

Add: Since the denominators are opposites, either one can serve as the LCD. If we choose x – y, we can multiply Y. by = to build it into an equivalent rational expression with the denominator x - y. When y - x is multiplied by -1, the subtraction is reversed, and the result is x - y. +Y y - x so that it has a denominator of x - y. + Build x - y Multiply the numerators. Multiply the denominators. %3D + X - y Rewrite the second denominator, -y + x, as x - y. The fractions now have a common denominator. x - y x - y Add the numerators. Write the difference over the common denominator x - y. x - y Simplify. 5a Add: a b – a

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Concept explainers

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.

Topic Video

Question

### Adding Rational Expressions with Opposite Denominators

#### Example Problem:
Add: $ \frac{x}{x-y} + \frac{y}{y-x} $

Since the denominators are opposites, either one can serve as the Least Common Denominator (LCD). If we choose $ x - y $, we can multiply $ \frac{y}{y - x} $ by $ \frac{-1}{-1} $ to build it into an equivalent rational expression with the denominator $ x - y $.

#### Step-by-Step Solution:
1. **Initial Expression**:
\[
\frac{x}{x-y} + \frac{y}{y-x}
\]

2. **Equating Denominators**:
Since the denominators are opposites,
\[
\frac{x}{x-y} + \frac{y}{y-x} = \frac{x}{x-y} + \frac{y}{y-x} \cdot \frac{-1}{-1}
\]

3. **Multiplication**:
\[
\frac{x}{x-y} + \frac{y \cdot (-1)}{(y-x) \cdot (-1)}
\]

4. **Rewrite Denominator**:
Rewrite $ -y + x $ as $ x - y $:
\[
\frac{x}{x-y} + \frac{-y}{x-y}
\]

5. **Combining the Numerators**:
Add the numerators while keeping the common denominator:
\[
\frac{x + (-y)}{x - y}
\]

6. **Simplify**:
\[
\frac{x - y}{x - y} = 1
\]

The final simplified result is $ 1 $.

#### Another Problem to Solve:
Add: $ \frac{5a}{a-b} + \frac{b}{b-a} $

Complete this problem using the same steps outlined above. Don’t forget to rewrite the denominators to a common term before combining the numerators and simplifying.

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