Solving Math Equations Step By Step Calculating (-1) + 5 - (-6) - 5

Let's break down this mathematical expression and find the correct answer. The expression we're working with is $(-1) + 5 - (-6) - 5$. This might seem a bit tricky at first glance, but by following the order of operations and understanding how to handle negative numbers, we can solve it easily.

Understanding the Basics

Before we dive into the solution, let's quickly review some fundamental concepts. First, remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this case, we mainly deal with addition and subtraction. Second, remember the rules for dealing with negative numbers:

• Adding a positive number is straightforward: $5 + 3 = 8$
• Adding a negative number is the same as subtracting: $5 + (-3) = 5 - 3 = 2$
• Subtracting a positive number is also straightforward: $5 - 3 = 2$
• Subtracting a negative number is the same as adding: $5 - (-3) = 5 + 3 = 8$. This is a crucial rule that we'll use in our problem.

With these basics in mind, let's tackle the expression step by step.

Step-by-Step Solution

1. Start with the first two terms: $(-1) + 5$

   Think of this as starting at -1 on the number line and moving 5 units to the right. This gives us:

   $(-1) + 5 = 4$

2. Next, we have $4 - (-6)$: This is where the rule of subtracting a negative comes into play.

   Subtracting a negative number is the same as adding its positive counterpart:

   $4 - (-6) = 4 + 6 = 10$

3. Finally, we subtract 5 from the result: $10 - 5$

   This is a simple subtraction:

   $10 - 5 = 5$

So, putting it all together:

$(-1) + 5 - (-6) - 5 = 4 - (-6) - 5 = 10 - 5 = 5$

Detailed Breakdown of Each Step

To ensure clarity, let's delve deeper into each step of the calculation. This detailed explanation will reinforce your understanding and help you tackle similar problems with confidence.

1. Adding a Negative to a Positive: $(-1) + 5$

When we encounter $(-1) + 5$, it's crucial to visualize this on a number line. Imagine starting at -1 and moving five units to the right. Each unit moved represents an addition of 1. Therefore, moving one unit from -1 brings us to 0, and moving the remaining four units lands us at 4. Mathematically, this operation can be seen as the difference between the absolute values of the numbers, with the sign being the same as the number with the larger absolute value. In this case, the absolute value of 5 is greater than the absolute value of -1, so the result is positive. Thus, $(-1) + 5 = 4$.

2. Subtracting a Negative: $4 - (-6)$

This is a critical concept in dealing with signed numbers. The operation $4 - (-6)$ might initially appear complex, but it simplifies significantly when we remember that subtracting a negative number is equivalent to adding its positive counterpart. Think of it as removing a debt. If you remove a debt of 6, it's the same as gaining 6. Thus, $4 - (-6)$ becomes $4 + 6$. This transformation makes the calculation straightforward: $4 + 6 = 10$. This step is a cornerstone in simplifying expressions with negative numbers, making it easier to arrive at the correct answer.

3. Final Subtraction: $10 - 5$

The last step in our calculation is $10 - 5$. This is a basic subtraction operation. We start with 10 and subtract 5, which leaves us with 5. This step is a straightforward application of arithmetic and solidifies the final result. There are no complexities here, just a simple deduction that brings us to the solution.

Why This Works: The Number Line Visualization

A number line is a powerful tool for understanding operations with negative numbers. It provides a visual representation of addition and subtraction. Let's revisit our steps using the number line:

• For $(-1) + 5$, start at -1 and move 5 units to the right, landing at 4.
• For $4 - (-6)$, remember that subtracting a negative is the same as adding. So, from 4, move 6 units to the right, landing at 10.
• For $10 - 5$, start at 10 and move 5 units to the left, landing at 5.

This visualization reinforces the logic behind each step and helps prevent common errors.

Common Mistakes to Avoid

When dealing with expressions like this, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

1. Forgetting the Rule of Subtracting a Negative: One of the most frequent errors is not recognizing that subtracting a negative number is the same as adding. For example, mistaking $4 - (-6)$ for $4 - 6$ can lead to an incorrect answer.
2. Incorrect Order of Operations: While this expression primarily involves addition and subtraction (which are performed from left to right), it’s essential to always keep the order of operations in mind for more complex expressions.
3. Sign Errors: Pay close attention to the signs of the numbers. A simple sign error can throw off the entire calculation. For instance, if you incorrectly calculate $(-1) + 5$ as -4 instead of 4, the subsequent steps will also be wrong.
4. Rushing Through the Calculation: It’s easy to make mistakes when you rush. Take your time, write out each step, and double-check your work.

By being mindful of these potential errors and taking a methodical approach, you can increase your accuracy and confidence in solving mathematical expressions.

Practice Problems

To solidify your understanding, try solving these similar problems:

1. $(-3) + 7 - (-2) - 4$
2. $2 - (-5) + (-8) - 1$
3. $(-6) + 4 - (-3) + 2$

Work through these problems step by step, applying the principles we've discussed. Check your answers to ensure you're on the right track. The more you practice, the more comfortable you'll become with these types of calculations.

Real-World Applications

Understanding how to work with negative numbers isn't just a mathematical exercise; it has practical applications in everyday life. Consider these scenarios:

1. Finance: When managing bank accounts, you deal with both positive balances (credits) and negative balances (debits). Calculating your account balance often involves adding and subtracting negative numbers.
2. Temperature: Temperatures can drop below zero, especially in cold climates. If the temperature is -5°C and rises by 10°C, the new temperature is -5 + 10 = 5°C.
3. Altitude: In geography, sea level is considered zero altitude. Locations below sea level have negative altitudes. For example, the Dead Sea has an altitude of about -430 meters.
4. Sports: In some sports, such as golf, scores can be positive or negative relative to par. A score of -2 means two strokes under par.

By mastering the basics of operations with negative numbers, you're not just improving your math skills; you're also enhancing your ability to understand and navigate real-world situations.

Conclusion

The correct answer to the expression $(-1) + 5 - (-6) - 5$ is 5. We arrived at this answer by carefully applying the rules of addition and subtraction, particularly the rule that subtracting a negative number is the same as adding its positive counterpart. By breaking down the problem into manageable steps and visualizing the operations on a number line, we can solve even seemingly complex expressions with confidence. Remember to avoid common mistakes, practice regularly, and apply these skills to real-world scenarios. With dedication and a clear understanding of the principles, you'll master the art of working with negative numbers.