Simplify The Expression: $ X - \frac{x^2}{x+y} }$Choose The Correct Simplified Form A. { \frac{-x^2 {x+y}$}$B. { \frac{xy}{x+y}$}$C. { \frac{2x^2+xy}{x+y}$}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. In this article, we will focus on simplifying a specific expression involving variables and fractions. We will break down the expression step by step, using algebraic manipulations to arrive at the simplified form.

The Expression to Simplify

The given expression is:

x−x2x+yx - \frac{x^2}{x+y}

Our goal is to simplify this expression and choose the correct simplified form from the options provided.

Step 1: Identify Common Factors

To simplify the expression, we need to identify any common factors that can be canceled out. In this case, we can see that both terms have a common factor of xx.

x - \frac{x^2}{x+y} = x\left(1 - \frac{x}{x+y}\right)

Step 2: Simplify the Fraction

Now, we can simplify the fraction inside the parentheses by finding a common denominator.

1 - \frac{x}{x+y} = \frac{x+y-x}{x+y} = \frac{y}{x+y}

Step 3: Combine the Terms

We can now combine the terms by multiplying the common factor xx with the simplified fraction.

x\left(\frac{y}{x+y}\right) = \frac{xy}{x+y}

Conclusion

Based on our step-by-step simplification, we have arrived at the simplified form of the expression:

xyx+y\frac{xy}{x+y}

This matches option B, which is the correct simplified form of the given expression.

Why is this the Correct Answer?

To understand why this is the correct answer, let's analyze the options provided.

  • Option A: −x2x+y\frac{-x^2}{x+y} is incorrect because it does not match the simplified form we arrived at.
  • Option C: 2x2+xyx+y\frac{2x^2+xy}{x+y} is also incorrect because it introduces an additional term that is not present in the original expression.
  • Option B: xyx+y\frac{xy}{x+y} is the correct answer because it matches the simplified form we arrived at through our step-by-step process.

Tips and Tricks

When simplifying expressions, it's essential to:

  • Identify common factors and cancel them out
  • Simplify fractions by finding a common denominator
  • Combine terms by multiplying or adding like terms
  • Check your work by plugging in values or using algebraic manipulations

By following these tips and tricks, you can simplify expressions efficiently and accurately.

Common Mistakes to Avoid

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

  • Failing to identify common factors or cancel them out
  • Simplifying fractions incorrectly or not finding a common denominator
  • Combining terms incorrectly or not checking your work
  • Not using algebraic manipulations or plugging in values to verify your answer

By being aware of these common mistakes, you can avoid them and simplify expressions accurately.

Conclusion

Introduction

In our previous article, we simplified the expression x−x2x+yx - \frac{x^2}{x+y} and arrived at the correct simplified form: xyx+y\frac{xy}{x+y}. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

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

A: The first step in simplifying an expression is to identify any common factors that can be canceled out. This involves looking for terms that have a common factor and canceling them out to simplify the expression.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find a common denominator. This involves multiplying the numerator and denominator by the same value to eliminate any common factors.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing the expression to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables. However, you need to be careful when simplifying expressions with multiple variables, as the order of operations may affect the final result.

Q: How do I know if I have simplified an expression correctly?

A: To ensure that you have simplified an expression correctly, you can plug in values or use algebraic manipulations to verify your answer. This will help you identify any errors or mistakes in your simplification.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

  • Failing to identify common factors or cancel them out
  • Simplifying fractions incorrectly or not finding a common denominator
  • Combining terms incorrectly or not checking your work
  • Not using algebraic manipulations or plugging in values to verify your answer

Q: Can I use a calculator to simplify expressions?

A: While calculators can be useful for simplifying expressions, it's essential to understand the underlying math and be able to simplify expressions manually. This will help you develop problem-solving skills and avoid relying on technology.

Q: How do I simplify expressions with negative numbers?

A: To simplify expressions with negative numbers, you need to follow the same steps as simplifying expressions with positive numbers. However, you need to be careful when dealing with negative numbers, as they can affect the final result.

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 simplifying expressions with variables in the denominator, as the order of operations may affect the final result.

Conclusion

Simplifying expressions is a crucial skill in mathematics that helps us solve problems efficiently and accurately. By following a step-by-step approach, identifying common factors, simplifying fractions, and combining terms, we can arrive at the simplified form of an expression. In this article, we answered some frequently asked questions related to simplifying expressions and provided tips and tricks to help you simplify expressions accurately and efficiently.

Additional Resources

For more information on simplifying expressions, check out the following resources:

  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

Practice Problems

To practice simplifying expressions, try the following problems:

  • Simplify the expression: x−x2x+yx - \frac{x^2}{x+y}
  • Simplify the expression: 2x2+xyx+y\frac{2x^2+xy}{x+y}
  • Simplify the expression: x−2x2x+yx - \frac{2x^2}{x+y}

By practicing simplifying expressions, you can develop your problem-solving skills and become more confident in your math abilities.