Simplifying The Expression: $-5-(-23)+(-13)-19$

by Alex Johnson 48 views

Let's dive into simplifying this numerical expression step-by-step. Simplifying mathematical expressions is a fundamental skill in algebra and arithmetic. In this comprehensive guide, we'll break down each part of the problem, ensuring you understand the process thoroughly. We'll tackle negative numbers, subtraction, and addition, providing a clear path to the solution. This article aims to help you grasp the underlying concepts, so you can confidently solve similar problems in the future. So, grab your pen and paper, and let’s get started!

Understanding the Basics

Before we jump into the actual calculation, let's quickly recap some essential rules for dealing with negative numbers and arithmetic operations. Understanding negative numbers is crucial here. Remember, subtracting a negative number is the same as adding its positive counterpart. For instance, −(−23)-(-23) becomes +23+23. Additionally, adding a negative number is the same as subtracting its positive value; (+(−13))(+(-13)) is the same as −13-13. With these basics in mind, we're ready to simplify the expression.

Step-by-Step Simplification

Now, let's tackle the expression −5−(−23)+(−13)−19-5-(-23)+(-13)-19 step by step. Our main keyword here is simplifying expressions, and we will do just that.

  1. Rewrite the Expression: First, let's rewrite the expression to make it easier to understand. We'll address the double negatives and the addition of the negative number:

    −5−(−23)+(−13)−19=−5+23−13−19-5-(-23)+(-13)-19 = -5 + 23 - 13 - 19

    This transformation makes the expression less cluttered and easier to work with.

  2. Combine the First Two Terms: Next, let's combine the first two terms, −5-5 and +23+23:

    −5+23=18-5 + 23 = 18

    So, our expression now looks like this:

    18−13−1918 - 13 - 19

  3. Combine the Next Two Terms: Now, let's subtract 1313 from 1818:

    18−13=518 - 13 = 5

    Our expression is further simplified to:

    5−195 - 19

  4. Final Subtraction: Finally, let's subtract 1919 from 55:

    5−19=−145 - 19 = -14

    Thus, the simplified value of the expression is −14-14.

Detailed Breakdown of Each Step

To ensure clarity, let's break down each step with a bit more detail. Detailed step breakdowns help to reinforce the process. We started with the expression:

−5−(−23)+(−13)−19-5-(-23)+(-13)-19

The first critical step is handling the subtraction of a negative number. Subtracting a negative is the same as adding a positive. So, −(−23)-(-23) becomes +23+23. Similarly, adding a negative number is the same as subtracting its positive counterpart. Hence, +(−13)+(-13) becomes −13-13. Rewriting the expression gives us:

−5+23−13−19-5 + 23 - 13 - 19

Now, we perform the operations from left to right. Adding −5-5 and 2323:

−5+23=18-5 + 23 = 18

So the expression becomes:

18−13−1918 - 13 - 19

Next, subtract 1313 from 1818:

18−13=518 - 13 = 5

Now the expression simplifies to:

5−195 - 19

Finally, subtracting 1919 from 55 gives us:

5−19=−145 - 19 = -14

Therefore, the simplified form of the original expression is −14-14.

Common Mistakes to Avoid

When simplifying numerical expressions, it's easy to make mistakes if you're not careful. Let's look at some common pitfalls to avoid.

  • Incorrectly Handling Negative Signs: One of the most common errors is mishandling negative signs. Remember, subtracting a negative number is the same as adding a positive number. Forgetting this rule can lead to significant errors. Always double-check how you are handling negative signs, especially when multiple negatives are involved.

  • Order of Operations: While this particular expression doesn't involve different operations (like multiplication or division), it’s crucial to remember the order of operations (PEMDAS/BODMAS) in more complex expressions. Make sure to perform operations in the correct order to get the right answer.

  • Arithmetic Errors: Simple arithmetic mistakes can also throw off your calculations. Take your time and double-check each step to ensure accuracy. It's easy to make a small mistake when adding or subtracting, so paying close attention can save you from errors.

Practice Problems

To solidify your understanding, let's try a few practice problems. Practice problems are key to mastering simplification. Here are some similar expressions for you to simplify:

  1. −10−(−15)+(−7)−3-10 - (-15) + (-7) - 3
  2. 8−(−4)+(−12)−68 - (-4) + (-12) - 6
  3. −3−(−9)+(−5)−11-3 - (-9) + (-5) - 11

Work through each problem step-by-step, following the methods we discussed. Check your answers to ensure you’re on the right track.

Solutions to Practice Problems

Let's go through the solutions to the practice problems to ensure you understand each step.

  1. Problem: −10−(−15)+(−7)−3-10 - (-15) + (-7) - 3

    • Rewrite: −10+15−7−3-10 + 15 - 7 - 3
    • Combine −10+15=5-10 + 15 = 5
    • Expression: 5−7−35 - 7 - 3
    • Combine 5−7=−25 - 7 = -2
    • Final: −2−3=−5-2 - 3 = -5

    Solution: -5

  2. Problem: 8−(−4)+(−12)−68 - (-4) + (-12) - 6

    • Rewrite: 8+4−12−68 + 4 - 12 - 6
    • Combine 8+4=128 + 4 = 12
    • Expression: 12−12−612 - 12 - 6
    • Combine 12−12=012 - 12 = 0
    • Final: 0−6=−60 - 6 = -6

    Solution: -6

  3. Problem: −3−(−9)+(−5)−11-3 - (-9) + (-5) - 11

    • Rewrite: −3+9−5−11-3 + 9 - 5 - 11
    • Combine −3+9=6-3 + 9 = 6
    • Expression: 6−5−116 - 5 - 11
    • Combine 6−5=16 - 5 = 1
    • Final: 1−11=−101 - 11 = -10

    Solution: -10

Real-World Applications

Understanding how to simplify expressions isn't just an abstract math skill. It has numerous real-world applications. Whether you're managing your finances, calculating distances, or even cooking, simplifying expressions can come in handy. For example, if you're calculating your monthly budget, you might need to add and subtract various expenses and income, some of which might be represented as negative numbers (like debt). Similarly, in physics, you might use these skills to calculate net forces or changes in energy. The ability to simplify expressions accurately is a valuable skill in many areas of life.

Conclusion

Simplifying the expression −5−(−23)+(−13)−19-5-(-23)+(-13)-19 involves understanding how to handle negative numbers and perform arithmetic operations in the correct order. By breaking down the problem step-by-step, we found that the simplified value is −14-14. Remember to pay close attention to negative signs and practice regularly to master these skills. With a solid understanding of these principles, you'll be well-equipped to tackle more complex mathematical problems. Keep practicing, and you’ll find that simplifying expressions becomes second nature!

For further learning and practice on similar math concepts, consider visiting Khan Academy, a trusted resource for math education.