Evaluating The Expression -5 + (-13) + 10 - 10 A Step-by-Step Guide

Evaluating mathematical expressions is a fundamental skill in algebra and arithmetic. In this article, we will thoroughly evaluate the expression -5 + (-13) + 10 - 10. This particular expression involves a combination of addition and subtraction with both positive and negative integers. We will break down each step methodically to ensure clarity and accuracy in our calculation. By understanding the basic rules of integer arithmetic, you can confidently solve similar expressions and build a strong foundation in mathematics. The steps involved in simplifying this expression highlight the importance of following the order of operations and the rules for adding and subtracting signed numbers. Let's dive in and explore how to correctly evaluate this expression, making sure each step is clear and easy to follow.

Understanding the Basics of Integer Arithmetic

Before we dive into the specifics of the expression -5 + (-13) + 10 - 10, it’s crucial to have a solid grasp of the basic principles of integer arithmetic. Integers are whole numbers, which can be positive, negative, or zero. Understanding how to add and subtract these numbers is key to solving more complex mathematical problems. When adding integers with the same sign, you simply add their absolute values and keep the sign. For example, adding two positive integers like 3 + 5 results in 8, and adding two negative integers like -4 + (-2) results in -6. The process changes slightly when adding integers with different signs. In this case, you subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value. For instance, to solve -7 + 3, you subtract 3 from 7 (which gives you 4) and since -7 has a larger absolute value, the answer is -4. Subtraction can be thought of as adding the opposite. For example, a - b is the same as a + (-b). This transformation is crucial when dealing with a mix of additions and subtractions, as it allows us to treat every operation as an addition, simplifying the overall calculation. These fundamental concepts form the backbone of integer arithmetic and are essential for accurately evaluating expressions like the one we're focusing on today. Mastering these basics will not only help in solving mathematical problems but also in everyday situations where you need to deal with positive and negative numbers.

Step-by-Step Evaluation of the Expression

Let's begin the step-by-step evaluation of the expression -5 + (-13) + 10 - 10. The initial step involves simplifying the addition of the negative numbers. We have -5 plus -13, which can be rewritten as -5 + (-13). When adding two negative numbers, we add their absolute values and keep the negative sign. So, the absolute value of -5 is 5, and the absolute value of -13 is 13. Adding these gives us 5 + 13 = 18. Since both numbers are negative, the result is -18. Now our expression looks like this: -18 + 10 - 10. The next step is to address the addition of -18 and 10. Here, we are adding integers with different signs, so we subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. The absolute value of -18 is 18, and the absolute value of 10 is 10. Subtracting 10 from 18 gives us 8. Since -18 has a larger absolute value, the result will be negative, so we have -8. Now our expression is simplified to -8 - 10. The final step is to handle the subtraction. Subtracting 10 from -8 is the same as adding -10 to -8. We again have two negative numbers, so we add their absolute values and keep the negative sign. The absolute value of -8 is 8, and the absolute value of -10 is 10. Adding these gives us 8 + 10 = 18. Since both numbers are negative, the final result is -18. Thus, the step-by-step evaluation of the expression clearly shows that -5 + (-13) + 10 - 10 = -18. Each step is crucial in reaching the correct answer, highlighting the importance of following the rules of integer arithmetic.

Detailed Breakdown of Each Operation

To further clarify the evaluation of the expression -5 + (-13) + 10 - 10, let’s provide a detailed breakdown of each operation. This in-depth look will help solidify understanding and prevent common errors. First, we tackle the addition of -5 and -13. This operation, -5 + (-13), involves adding two negative integers. As we discussed, when adding two negative numbers, we add their absolute values and retain the negative sign. The absolute value of -5 is 5, and the absolute value of -13 is 13. Adding these together, we get 5 + 13 = 18. Since both numbers are negative, the result is -18. This step simplifies the expression to -18 + 10 - 10. Next, we address the addition of -18 and 10. Here, we are adding integers with different signs. When adding integers with different signs, we subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value. The absolute value of -18 is 18, and the absolute value of 10 is 10. Subtracting 10 from 18 gives us 8. Since -18 has a larger absolute value, the result will be negative, so we have -8. Now, the expression becomes -8 - 10. Finally, we handle the subtraction of 10 from -8. Subtracting a number is the same as adding its opposite, so -8 - 10 is equivalent to -8 + (-10). We are now adding two negative numbers, so we add their absolute values and keep the negative sign. The absolute value of -8 is 8, and the absolute value of -10 is 10. Adding these, we get 8 + 10 = 18. Since both numbers are negative, the result is -18. Therefore, the final answer is -18. By breaking down each operation and explaining the underlying principles, we ensure a comprehensive understanding of how to evaluate the expression.

Common Mistakes and How to Avoid Them

When evaluating expressions like -5 + (-13) + 10 - 10, it’s easy to make common mistakes if one isn't careful. Identifying these pitfalls and understanding how to avoid them is crucial for achieving accurate results. One of the most frequent errors is misapplying the rules for adding and subtracting negative numbers. For example, some might incorrectly subtract the absolute values of -5 and -13 when they should be adding them. Remember, when adding two negative numbers, you add their absolute values and keep the negative sign. Another common mistake occurs when dealing with subtraction. Students sometimes forget that subtracting a number is the same as adding its opposite. For instance, -8 - 10 should be treated as -8 + (-10), which is adding two negative numbers. Failing to make this transformation can lead to incorrect calculations. Order of operations, though seemingly straightforward, is another area where mistakes can happen. In this particular expression, addition and subtraction are performed from left to right, but in more complex expressions, not following the correct order (PEMDAS/BODMAS) can result in errors. Careless arithmetic is also a frequent cause of mistakes. It’s essential to double-check each step to ensure accuracy. A simple miscalculation in one step can propagate through the entire problem, leading to a wrong final answer. To avoid these mistakes, it’s helpful to practice breaking down each problem into smaller, manageable steps. Write each step clearly, showing your work as you go. This not only helps you keep track of your calculations but also makes it easier to identify any errors. Regular practice and a thorough understanding of the rules of integer arithmetic are the best ways to minimize mistakes and build confidence in your mathematical abilities. By being mindful of these common pitfalls, you can improve your accuracy and approach similar problems with greater assurance.

Conclusion: The Final Result and Its Significance

In conclusion, after a meticulous step-by-step evaluation, we have determined that the expression -5 + (-13) + 10 - 10 equals -18. This final result is not just a numerical answer but a testament to the importance of understanding and applying the fundamental rules of integer arithmetic. The process we followed, from simplifying the initial expression to breaking down each operation, highlights the necessity of precision and attention to detail in mathematics. Evaluating expressions like this is more than just a classroom exercise; it's a foundational skill that extends to various aspects of mathematics and everyday problem-solving. The ability to confidently handle integers, whether positive or negative, is crucial for more advanced mathematical concepts such as algebra, calculus, and beyond. Moreover, the skills developed in solving such expressions—logical thinking, step-by-step problem-solving, and error checking—are valuable in numerous fields and life situations. The significance of correctly evaluating the expression lies not only in obtaining the correct answer but also in reinforcing mathematical principles that empower you to tackle more complex challenges. By understanding the underlying concepts and avoiding common pitfalls, you can build a solid foundation in mathematics and enhance your problem-solving abilities. Therefore, mastering the evaluation of expressions like -5 + (-13) + 10 - 10 is a crucial step towards mathematical proficiency and overall analytical thinking.