Simplify The Expression:${ \frac{x}{x-y} - \frac{x}{x+y} + \frac{2xy}{x 2+y 2} }$

Mar 11, 2025 by ADMIN 83 views

Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying a given expression involving fractions and variables. The expression is $\frac{x}{x-y} - \frac{x}{x+y} + \frac{2xy}{x^2+y^2}$ , and we will break it down step by step to arrive at a simplified form.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what it represents. The expression consists of three fractions, each with a different denominator. The first two fractions have denominators $(x-y)$ and $(x+y)$ , respectively, while the third fraction has a denominator $(x^2+y^2)$ . The numerators of the first two fractions are both $x$ , while the numerator of the third fraction is $2xy$ .

Simplifying the Expression

To simplify the expression, we can start by finding a common denominator for the first two fractions. The common denominator is $(x-y)(x+y)$ , which can be expanded to $x^2-y^2$ . We can rewrite the first two fractions with this common denominator:

\frac{x}{x-y} - \frac{x}{x+y} = \frac{x(x+y)}{(x-y)(x+y)} - \frac{x(x-y)}{(x-y)(x+y)}

Combining the Fractions

Now that we have a common denominator, we can combine the two fractions by adding or subtracting their numerators:

\frac{x(x+y)}{(x-y)(x+y)} - \frac{x(x-y)}{(x-y)(x+y)} = \frac{x(x+y) - x(x-y)}{(x-y)(x+y)}

Expanding the Numerator

We can expand the numerator by distributing the negative sign:

x(x+y) - x(x-y) = x^2 + xy - x^2 + xy

Canceling Out Terms

Now that we have expanded the numerator, we can cancel out the common terms:

x^2 + xy - x^2 + xy = 2xy

Adding the Third Fraction

Now that we have simplified the first two fractions, we can add the third fraction to the expression:

\frac{2xy}{x^2+y^2} + \frac{2xy}{x^2+y^2}

Combining the Fractions

We can combine the two fractions by adding their numerators:

\frac{2xy}{x^2+y^2} + \frac{2xy}{x^2+y^2} = \frac{4xy}{x^2+y^2}

Final Simplification

The final simplified expression is $\frac{4xy}{x^2+y^2}$ .

Conclusion

In this article, we simplified a given expression involving fractions and variables. We started by finding a common denominator for the first two fractions, then combined them by adding or subtracting their numerators. We expanded the numerator, canceled out common terms, and added the third fraction to the expression. Finally, we combined the fractions and arrived at the final simplified expression.

Tips and Tricks

When simplifying expressions, it's essential to find a common denominator for fractions with different denominators.
Combining fractions by adding or subtracting their numerators can help simplify the expression.
Expanding the numerator and canceling out common terms can also help simplify the expression.
Adding fractions with the same denominator is straightforward, but adding fractions with different denominators requires finding a common denominator.

Real-World Applications

Simplifying expressions is a fundamental skill in mathematics, and it has numerous real-world applications. In physics, for example, simplifying expressions can help solve problems involving motion, energy, and momentum. In engineering, simplifying expressions can help design and optimize systems, such as electrical circuits and mechanical systems.

Common Mistakes

When simplifying expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

Failing to find a common denominator for fractions with different denominators.
Not combining fractions by adding or subtracting their numerators.
Not expanding the numerator and canceling out common terms.
Not adding fractions with the same denominator.

Final Thoughts

Simplifying expressions is a crucial skill in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you can simplify expressions involving fractions and variables. Remember to find a common denominator, combine fractions by adding or subtracting their numerators, expand the numerator, and cancel out common terms. With practice and patience, you can become proficient in simplifying expressions and tackle even the most complex problems.

Introduction

In our previous article, we simplified a given expression involving fractions and variables. We broke down the expression step by step, finding a common denominator, combining fractions, expanding the numerator, and canceling out common terms. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to find a common denominator for fractions with different denominators.

Q: How do I find a common denominator?

A: To find a common denominator, you can multiply the denominators of the fractions together. For example, if you have two fractions with denominators $(x-y)$ and $(x+y)$ , the common denominator would be $(x-y)(x+y)$ .

Q: What if I have multiple fractions with different denominators?

A: If you have multiple fractions with different denominators, you can find a common denominator by multiplying all the denominators together. For example, if you have three fractions with denominators $(x-y)$ , $(x+y)$ , and $(x^2+y^2)$ , the common denominator would be $(x-y)(x+y)(x^2+y^2)$ .

Q: How do I combine fractions with the same denominator?

A: Combining fractions with the same denominator is straightforward. You can simply add or subtract the numerators of the fractions.

Q: What if I have fractions with different signs?

A: If you have fractions with different signs, you can combine them by adding or subtracting their numerators. For example, if you have two fractions with numerators $x$ and $-x$ , you can combine them by adding their numerators: $x + (-x) = 0$ .

Q: Can I simplify expressions with variables in the denominator?

A: Yes, you can simplify expressions with variables in the denominator. However, you need to be careful when canceling out common terms. For example, if you have an expression with a denominator of $(x-y)$ and a numerator of $x$ , you can cancel out the common term $x$ by dividing both the numerator and the denominator by $x$ .

Q: What if I have an expression with a zero denominator?

A: If you have an expression with a zero denominator, it is undefined. You cannot simplify the expression in this case.

Q: Can I simplify expressions with multiple variables?

A: Yes, you can simplify expressions with multiple variables. However, you need to be careful when canceling out common terms. For example, if you have an expression with a denominator of $(x-y)$ and a numerator of $x+y$ , you can cancel out the common term $x$ by dividing both the numerator and the denominator by $x$ .

Q: What if I have an expression with a fraction in the numerator?

A: If you have an expression with a fraction in the numerator, you can simplify it by finding a common denominator for the fraction in the numerator and the fraction in the denominator.

Q: Can I simplify expressions with complex numbers?

A: Yes, you can simplify expressions with complex numbers. However, you need to be careful when canceling out common terms. For example, if you have an expression with a denominator of $(x-y)$ and a numerator of $x+iy$ , you can cancel out the common term $x$ by dividing both the numerator and the denominator by $x$ .