Simplifying Numerical Expressions 35 + (-13) + (+8) - (-6) Step-by-Step

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In the realm of mathematics, simplifying numerical expressions is a fundamental skill. This article will meticulously guide you through the process of simplifying the expression 35 + (-13) + (+8) - (-6), elucidating each step to ensure clarity and comprehension. By the end of this guide, you'll not only understand how to solve this specific problem but also grasp the underlying principles applicable to a wide array of similar expressions. Our main focus revolves around the core arithmetic operations – addition and subtraction – and how they interact with positive and negative numbers. Understanding these interactions is crucial for mastering not just basic arithmetic but also more advanced mathematical concepts. The journey to simplifying this expression involves understanding the rules of integer arithmetic, particularly how to handle the addition and subtraction of positive and negative numbers. Let's embark on this mathematical journey together, breaking down the expression into manageable steps and unveiling the correct solution. Before diving into the step-by-step simplification, let's briefly discuss the importance of order of operations. While this particular expression doesn't involve multiplication or division, it's worth noting that, in general, we follow the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). However, in this case, we primarily deal with addition and subtraction, which are performed from left to right. This foundational understanding will pave the way for a seamless simplification process.

Breaking Down the Expression

Let's begin by restating the expression: 35 + (-13) + (+8) - (-6). To simplify this, we will proceed step-by-step, adhering to the order of operations, which, in this case, involves performing addition and subtraction from left to right. This approach ensures that we tackle the expression systematically, minimizing the chances of errors and maximizing clarity. The first part of the expression we encounter is 35 + (-13). Adding a negative number is equivalent to subtraction. So, 35 + (-13) becomes 35 - 13. This is a straightforward subtraction operation. Understanding this equivalence is crucial for handling expressions involving both positive and negative numbers. Next, we move to the second operation, which involves the result of the first operation and the next term in the expression. This sequential approach is key to simplifying complex expressions. As we progress, we will continue to highlight the underlying principles of integer arithmetic, such as the rules for adding and subtracting numbers with different signs. These principles form the bedrock of mathematical calculations and are essential for proficiency in algebra and beyond. By meticulously working through each step, we aim to provide not just a solution but a deep understanding of the process involved.

Step-by-Step Simplification

  1. First Operation: 35 + (-13)

    • Adding a negative number is the same as subtracting its positive counterpart. Therefore, 35 + (-13) is equivalent to 35 - 13.
    • 35 - 13 = 22

    The initial step involves the addition of a positive number and a negative number. Recognizing that adding a negative number is equivalent to subtraction is a fundamental concept in arithmetic. This transformation simplifies the expression and allows us to proceed with a standard subtraction operation. The result of this first operation, 22, becomes the starting point for the next step in our simplification journey. This methodical approach of breaking down the expression into smaller, manageable parts is crucial for accuracy and clarity. By focusing on one operation at a time, we minimize the risk of errors and ensure that each step is thoroughly understood. This approach is not only applicable to this specific problem but also serves as a valuable strategy for tackling more complex mathematical expressions in the future. The ability to confidently handle operations involving positive and negative numbers is a cornerstone of mathematical proficiency, and this step-by-step simplification reinforces that skill. Furthermore, it highlights the importance of precision and attention to detail in mathematical calculations.

  2. Second Operation: 22 + (+8)

    • Adding a positive number is straightforward addition.
    • 22 + 8 = 30

    Having simplified the first part of the expression, we now move on to the second operation: adding a positive number to our previous result. In this case, we are adding +8 to 22. This is a straightforward addition operation, where we simply combine the two positive numbers. The result of this operation, 30, further simplifies the expression and brings us closer to the final answer. This step reinforces the basic principles of addition and highlights the cumulative effect of each operation in the simplification process. By meticulously performing each addition and subtraction, we maintain accuracy and ensure that the final result is correct. The clarity of this step-by-step approach is particularly beneficial for learners, as it breaks down a seemingly complex expression into a series of simple operations. This method fosters a deeper understanding of the underlying mathematical concepts and builds confidence in handling numerical expressions. Moreover, it emphasizes the importance of careful calculation and attention to detail in achieving accurate results.

  3. Third Operation: 30 - (-6)

    • Subtracting a negative number is the same as adding its positive counterpart. Therefore, 30 - (-6) is equivalent to 30 + 6.
    • 30 + 6 = 36

    The final operation in simplifying the expression involves subtracting a negative number. This step is crucial as it often leads to errors if the rule is not clearly understood. Subtracting a negative number is mathematically equivalent to adding its positive counterpart. Therefore, 30 - (-6) transforms into 30 + 6. This transformation is a key concept in integer arithmetic and is essential for accurate calculations. Performing the addition, 30 + 6 equals 36. This final result represents the simplified value of the entire expression. This step not only provides the answer but also reinforces the importance of understanding the rules of signed number arithmetic. The ability to confidently handle subtraction of negative numbers is a fundamental skill in mathematics, and this example provides a clear illustration of its application. The step-by-step approach used throughout this simplification process ensures that each operation is performed correctly, leading to an accurate final result. Furthermore, it highlights the interconnectedness of mathematical concepts and the importance of mastering basic rules for tackling more complex problems.

Final Answer and Conclusion

Therefore, the simplified value of the expression 35 + (-13) + (+8) - (-6) is 36. This corresponds to option A. Through this detailed step-by-step simplification, we have demonstrated how to effectively handle expressions involving both positive and negative numbers. The key takeaway is the importance of understanding the rules of integer arithmetic, particularly how addition and subtraction interact with negative numbers. By following a systematic approach and breaking down the expression into smaller, manageable parts, we can minimize errors and arrive at the correct solution. This process not only solves the specific problem at hand but also equips us with the tools and knowledge to tackle a wider range of mathematical expressions. The ability to confidently simplify numerical expressions is a fundamental skill in mathematics, and this example serves as a valuable illustration of the principles involved. Furthermore, the emphasis on clarity and precision throughout the simplification process underscores the importance of these qualities in mathematical problem-solving. The journey from the initial expression to the final answer highlights the power of step-by-step reasoning and the cumulative effect of each operation. By mastering these techniques, students can build a strong foundation in mathematics and confidently approach more complex challenges.

Choosing the Correct Option

Based on our calculations, the correct answer is A) 36. The other options (B) 44, C) 50, and D) 24 are incorrect. This reinforces the importance of careful calculation and attention to detail when simplifying mathematical expressions. A single error in any step can lead to an incorrect final result. Therefore, it is crucial to double-check each operation and ensure that the rules of arithmetic are correctly applied. In this case, the step-by-step approach we followed allowed us to identify and avoid potential errors, leading us to the correct answer. The incorrect options may arise from common mistakes, such as misinterpreting the rules for adding or subtracting negative numbers, or making arithmetic errors during the calculations. By understanding the process thoroughly and practicing similar problems, one can minimize the likelihood of making such mistakes. This example serves as a reminder that accuracy and precision are paramount in mathematics, and that a systematic approach is often the best way to achieve the correct solution. Furthermore, it highlights the value of reviewing one's work to ensure that no errors have been made.

Practice Problems

To further solidify your understanding, try simplifying these expressions:

  1. 20 - (-5) + (-12) - 7
  2. -15 + 9 - (-4) + 2
  3. 18 + (-11) - 6 + (-3)

Working through these practice problems will provide valuable hands-on experience and help you develop fluency in simplifying numerical expressions. Remember to follow the same step-by-step approach we used in the example, paying close attention to the rules for adding and subtracting negative numbers. These practice problems are designed to reinforce the concepts discussed in this article and to provide an opportunity to apply your knowledge in different contexts. By tackling a variety of problems, you can build confidence in your abilities and develop a deeper understanding of the underlying mathematical principles. Furthermore, these exercises can help you identify any areas where you may need additional practice or clarification. The process of solving these problems will not only improve your arithmetic skills but also enhance your problem-solving abilities in general. Mathematics is a subject that builds upon itself, so mastering the fundamentals is essential for success in more advanced topics. These practice problems provide a solid foundation for further exploration of mathematical concepts.