Simplify The Expression:$\[ (5x^2y - 3xy + 25) - (2xy^2 + 6xy + 5) \\]

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying a given expression, which involves combining like terms and applying the rules of algebra. Our goal is to provide a clear and concise guide on how to simplify the expression, making it easier for readers to understand and apply the concepts.

The Given Expression

The given expression is:

{ (5x^2y - 3xy + 25) - (2xy^2 + 6xy + 5) \}

This expression consists of two sets of parentheses, each containing a combination of variables and constants. Our task is to simplify this expression by combining like terms and applying the rules of algebra.

Step 1: Distribute the Negative Sign

To simplify the expression, we need to start by distributing the negative sign to the terms inside the second set of parentheses. This will change the sign of each term inside the parentheses.

{ (5x^2y - 3xy + 25) - (2xy^2 + 6xy + 5) \}

{ (5x^2y - 3xy + 25) + (-2xy^2 - 6xy - 5) \}

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable(s) raised to the same power. In this case, we have two sets of like terms: the terms with $x^2y$, the terms with $xy$, and the constant terms.

{ (5x^2y - 3xy + 25) + (-2xy^2 - 6xy - 5) \}

{ 5x^2y - 3xy + 25 - 2xy^2 - 6xy - 5 \}

Step 3: Combine the Terms with $x^2y$

We can combine the terms with $x^2y$ by adding their coefficients.

{ 5x^2y - 3xy + 25 - 2xy^2 - 6xy - 5 \}

{ 5x^2y - 3xy - 6xy + 25 - 2xy^2 - 5 \}

Step 4: Combine the Terms with $xy$

We can combine the terms with $xy$ by adding their coefficients.

{ 5x^2y - 3xy - 6xy + 25 - 2xy^2 - 5 \}

{ 5x^2y - 9xy + 25 - 2xy^2 - 5 \}

Step 5: Combine the Constant Terms

We can combine the constant terms by adding them.

{ 5x^2y - 9xy + 25 - 2xy^2 - 5 \}

{ 5x^2y - 9xy + 20 - 2xy^2 \}

Step 6: Write the Final Answer

The final answer is:

{ 5x^2y - 9xy + 20 - 2xy^2 \}

Conclusion

Simplifying the given expression involved distributing the negative sign, combining like terms, and applying the rules of algebra. By following these steps, we were able to simplify the expression and arrive at the final answer. This guide provides a clear and concise explanation of the process, making it easier for readers to understand and apply the concepts.

Tips and Tricks

• When simplifying expressions, always start by distributing the negative sign to the terms inside the second set of parentheses.
• Combine like terms by adding their coefficients.
• Apply the rules of algebra, such as the distributive property and the commutative property.
• Check your work by plugging in values for the variables and evaluating the expression.

Real-World Applications

Simplifying expressions is a crucial skill that has numerous real-world applications. In mathematics, algebraic manipulation is used to solve equations and inequalities, which is essential in fields such as physics, engineering, and economics. In addition, simplifying expressions is also used in computer science, where it is used to optimize algorithms and improve the performance of computer programs.

Final Thoughts

Simplifying expressions is a fundamental concept in mathematics that requires practice and patience. By following the steps outlined in this guide, readers can develop the skills necessary to simplify complex expressions and apply the concepts to real-world problems. Whether you are a student or a professional, simplifying expressions is an essential skill that will serve you well in your future endeavors.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

Introduction

In our previous article, we provided a step-by-step guide on how to simplify a given expression. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will provide a Q&A guide to algebraic manipulation, focusing on simplifying expressions.

Q1: What is the first step in simplifying an expression?

A1: The first step in simplifying an expression is to distribute the negative sign to the terms inside the second set of parentheses. This will change the sign of each term inside the parentheses.

Q2: How do I combine like terms?

A2: To combine like terms, you need to add their coefficients. Like terms are terms that have the same variable(s) raised to the same power.

Q3: What are some common mistakes to avoid when simplifying expressions?

A3: Some common mistakes to avoid when simplifying expressions include:

• Not distributing the negative sign to the terms inside the second set of parentheses
• Not combining like terms
• Not applying the rules of algebra, such as the distributive property and the commutative property

Q4: How do I know if I have simplified an expression correctly?

A4: To check if you have simplified an expression correctly, you can plug in values for the variables and evaluate the expression. If the expression simplifies to the same value, then you have simplified it correctly.

Q5: What are some real-world applications of simplifying expressions?

A5: Simplifying expressions has numerous real-world applications, including:

• Solving equations and inequalities in physics, engineering, and economics
• Optimizing algorithms and improving the performance of computer programs in computer science
• Simplifying complex mathematical expressions in fields such as calculus and linear algebra

Q6: How can I practice simplifying expressions?

A6: You can practice simplifying expressions by:

• Working through practice problems in algebra textbooks or online resources
• Creating your own practice problems and simplifying them
• Using online tools or software to generate random expressions and simplify them

Q7: What are some common expressions that require simplification?

A7: Some common expressions that require simplification include:

• Expressions with multiple sets of parentheses
• Expressions with negative signs and variables
• Expressions with fractions and variables

Q8: How can I apply the rules of algebra to simplify expressions?

A8: To apply the rules of algebra to simplify expressions, you need to:

• Distribute the negative sign to the terms inside the second set of parentheses
• Combine like terms by adding their coefficients
• Apply the distributive property and the commutative property to simplify the expression

Q9: What is the importance of simplifying expressions in mathematics?

A9: Simplifying expressions is an essential skill in mathematics that allows you to:

• Solve equations and inequalities
• Optimize algorithms and improve the performance of computer programs
• Simplify complex mathematical expressions

Q10: How can I use technology to simplify expressions?

A10: You can use technology to simplify expressions by:

• Using online tools or software to generate random expressions and simplify them
• Using calculators or computer programs to simplify complex expressions
• Using graphing software to visualize and simplify expressions

Conclusion

Simplifying expressions is a fundamental concept in mathematics that requires practice and patience. By following the steps outlined in this Q&A guide, you can develop the skills necessary to simplify complex expressions and apply the concepts to real-world problems. Whether you are a student or a professional, simplifying expressions is an essential skill that will serve you well in your future endeavors.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.