Solving $(-5)+22+(-7)+8(-9)$ A Step-by-Step Guide

In this article, we will delve into the arithmetic problem $(-5) + 22 + (-7) + 8(-9)$, breaking it down step by step to ensure clarity and understanding. This seemingly simple equation encompasses fundamental mathematical operations, including addition, subtraction (through the addition of negative numbers), and multiplication. Mastering such problems is crucial for building a strong foundation in mathematics. We will explore the order of operations, the rules governing the addition and subtraction of integers, and the importance of careful calculation to arrive at the correct solution. So, let's embark on this mathematical journey together and unravel the intricacies of this equation.

Understanding the Order of Operations

At the heart of solving any arithmetic problem lies the understanding and application of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This hierarchical structure dictates the sequence in which mathematical operations should be performed to ensure consistency and accuracy in the final result. In our equation, $(-5) + 22 + (-7) + 8(-9)$, we encounter addition and multiplication. According to PEMDAS, multiplication takes precedence over addition. Therefore, our initial focus should be on the term $8(-9)$.

• The Primacy of Multiplication: The multiplication operation $8(-9)$ signifies multiplying the positive integer 8 by the negative integer -9. The rules governing the multiplication of signed numbers state that the product of a positive number and a negative number is always a negative number. In this case, $8 imes -9 = -72$. This crucial step transforms our original equation into $(-5) + 22 + (-7) + (-72)$, setting the stage for the subsequent addition and subtraction operations.
• Simplifying the Equation: By performing the multiplication first, we've effectively simplified the equation, reducing the complexity and making it easier to manage. This simplification is a testament to the power of adhering to the order of operations. It's a strategic move that prevents confusion and ensures that we proceed with a clear path towards the solution. The revised equation now presents a series of additions and subtractions, which we can tackle systematically from left to right.

By meticulously following the order of operations, we've navigated the initial hurdle and laid a solid groundwork for the remaining calculations. This methodical approach is not just a technique but a mindset – a commitment to precision and clarity in mathematical problem-solving.

Adding and Subtracting Integers

Now that we've simplified the equation to $(-5) + 22 + (-7) + (-72)$, we're faced with a series of additions and subtractions involving integers. Integers encompass both positive and negative whole numbers, including zero. Mastering the rules of adding and subtracting integers is paramount to accurately solving this equation. These rules dictate how to combine numbers with different signs, and understanding them is essential for achieving the correct final answer.

• The Dance of Signs: When adding integers with different signs, we essentially perform a subtraction. The magnitude of the result is determined by the difference between the absolute values of the numbers, and the sign of the result matches the sign of the integer with the larger absolute value. For instance, in the initial part of our equation, we have $(-5) + 22$. Here, we subtract the absolute value of -5 (which is 5) from the absolute value of 22 (which is 22), yielding 17. Since 22 has a larger absolute value and is positive, the result is positive 17. Thus, $(-5) + 22 = 17$.
• Continuing the Chain: We now have the simplified expression $17 + (-7) + (-72)$. Applying the same principle, we add 17 and -7. The difference between their absolute values is 10, and since 17 has a larger absolute value and is positive, the result is positive 10. Our equation further simplifies to $10 + (-72)$.
• The Final Countdown: Finally, we add 10 and -72. The difference between their absolute values is 62, and since -72 has a larger absolute value and is negative, the result is -62. Therefore, $10 + (-72) = -62$.

Through the systematic application of the rules governing the addition and subtraction of integers, we've meticulously navigated the equation, step by step, arriving at a single numerical value. This process highlights the importance of understanding the underlying principles that govern mathematical operations. It's not just about following steps; it's about comprehending why those steps lead to the correct solution.

Step-by-Step Solution and Detailed Explanation

To solidify our understanding, let's meticulously walk through the solution to the equation $(-5) + 22 + (-7) + 8(-9)$, providing a detailed explanation for each step. This comprehensive approach will not only reinforce the concepts we've discussed but also serve as a practical guide for tackling similar arithmetic problems in the future.

1. Tackling Multiplication First: As per the order of operations (PEMDAS), we begin with the multiplication operation: $8(-9)$. Multiplying 8 by -9 yields -72. This transforms our equation into $(-5) + 22 + (-7) + (-72)$. The multiplication step is a critical first move, as it simplifies the equation and prepares it for the subsequent addition and subtraction operations.
2. Adding -5 and 22: Next, we perform the addition $(-5) + 22$. Adding a negative number is equivalent to subtraction. So, we subtract the absolute value of -5 (which is 5) from 22, resulting in 17. The equation now becomes $17 + (-7) + (-72)$. This step demonstrates the interplay between addition and subtraction in the realm of integers.
3. Adding 17 and -7: Moving along, we add 17 and -7. Again, adding a negative number is akin to subtraction. Subtracting 7 from 17 gives us 10. Our equation further simplifies to $10 + (-72)$. This iterative process of simplification is a hallmark of effective problem-solving in mathematics.
4. The Final Summation: Finally, we add 10 and -72. Subtracting the absolute value of 10 from the absolute value of -72 (which is 72) gives us 62. Since -72 has a larger absolute value and is negative, the result is -62. Therefore, the final answer to the equation is -62.

• The Result: The solution to the arithmetic problem $(-5) + 22 + (-7) + 8(-9)$ is -62. This final answer is the culmination of a series of carefully executed operations, each guided by the principles of mathematical order and the rules governing integer arithmetic.

By breaking down the problem into manageable steps and providing a detailed explanation for each operation, we've not only solved the equation but also illuminated the underlying mathematical concepts. This step-by-step approach is invaluable for building confidence and proficiency in arithmetic problem-solving.

Common Mistakes and How to Avoid Them

Solving arithmetic problems like $(-5) + 22 + (-7) + 8(-9)$ can sometimes lead to errors if we're not careful. Understanding common mistakes and learning how to avoid them is crucial for achieving accuracy and building a strong foundation in mathematics. Let's explore some of these pitfalls and strategies for preventing them.

• Ignoring the Order of Operations: One of the most frequent errors is neglecting the order of operations (PEMDAS). Failing to perform multiplication before addition, for instance, can lead to a drastically incorrect answer. In our equation, if we were to add -5 and 22 before multiplying 8 and -9, we would be deviating from the fundamental rules of arithmetic. To avoid this, always remember PEMDAS and prioritize operations accordingly. Double-check your steps to ensure you're adhering to the correct sequence.
• Sign Errors with Integers: Another common source of mistakes is mishandling the signs of integers. Forgetting that adding a negative number is equivalent to subtraction, or incorrectly applying the rules for multiplying signed numbers, can result in errors. For example, mistaking $8(-9)$ for 72 instead of -72 would throw off the entire calculation. To prevent sign errors, it's essential to thoroughly understand the rules governing integer arithmetic. Take your time, and if necessary, write out the steps explicitly, paying close attention to the signs of each number.
• Calculation Mistakes: Simple arithmetic errors, such as miscalculating a sum or difference, can also occur. These seemingly minor mistakes can snowball and lead to an incorrect final answer. To minimize calculation errors, practice mental math, use a calculator for complex calculations, and always double-check your work. It's often helpful to solve the problem again from scratch to verify your result.
• Rushing Through the Process: In the rush to complete a problem, it's easy to skip steps or make careless errors. Haste can lead to mistakes that could have been easily avoided with a more deliberate approach. To avoid rushing, take your time, work methodically, and break the problem down into smaller, more manageable steps. This approach not only reduces the likelihood of errors but also enhances your understanding of the problem-solving process.

By being aware of these common pitfalls and actively employing strategies to avoid them, you can significantly improve your accuracy and confidence in solving arithmetic problems. Remember, precision and attention to detail are the cornerstones of mathematical success.

Conclusion: Mastering Arithmetic Operations

In conclusion, solving the arithmetic problem $(-5) + 22 + (-7) + 8(-9)$ is more than just arriving at the correct answer; it's a journey through the fundamental principles of mathematics. We've explored the critical role of the order of operations (PEMDAS), the intricacies of adding and subtracting integers, and the importance of meticulous step-by-step calculations. Through this process, we've not only determined that the solution is -62 but also gained a deeper understanding of the underlying mathematical concepts.

The ability to confidently and accurately solve such problems is a cornerstone of mathematical literacy. It's a skill that extends far beyond the classroom, finding application in everyday life, from managing personal finances to making informed decisions. By mastering these basic arithmetic operations, we empower ourselves to tackle more complex mathematical challenges with assurance and competence.

Furthermore, we've highlighted common mistakes that can occur when solving arithmetic problems and provided practical strategies for avoiding them. Recognizing the potential pitfalls and proactively addressing them is a key element of effective problem-solving. It's about not just getting the right answer but also developing the critical thinking skills necessary to navigate mathematical challenges successfully.

In essence, the journey through this equation has been a valuable exercise in mathematical reasoning and precision. It underscores the importance of a methodical approach, a thorough understanding of fundamental principles, and a commitment to accuracy. As we continue our exploration of mathematics, the lessons learned here will serve as a solid foundation for future endeavors. So, embrace the challenge, practice diligently, and watch your mathematical skills flourish.