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
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.
![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc60d0879-bf1d-4600-af9d-cdd728ad041f%2Fdf57b423-bc4a-47e2-a0a5-68b65c9dae98%2Fd6w2q8j_processed.png&w=3840&q=75)
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