Solving 8 + (-5) - (-6 * 3) + (7 - 10) A Step-by-Step Guide

Hey guys! Today, we're diving into a math problem that might look a little intimidating at first, but trust me, it's totally manageable when we break it down step by step. We're going to tackle the expression 8 + (-5) - (-6 * 3) + (7 - 10). Think of it like a puzzle – each part fits together, and we just need to find the right order to solve it. So, grab your pencils, and let's get started!

Understanding the Order of Operations

Before we even think about adding or subtracting, we need to remember our good old friend, the order of operations, often remembered by the acronym PEMDAS or BODMAS. This is the golden rule of math that tells us exactly what to do and when. It ensures everyone solves the same problem the same way, no matter who they are or where they're from.

Let's quickly break down what each letter stands for:

• Parentheses (or Brackets): First, we deal with anything inside parentheses or brackets.
• Exponents (or Orders): Next up are exponents or orders, like squares and cubes.
• Multiplication and Division: After that, we handle multiplication and division, working from left to right.
• Addition and Subtraction: Finally, we tackle addition and subtraction, also working from left to right.

This order is super crucial. Imagine if we just went from left to right without considering PEMDAS/BODMAS – we'd get a completely different answer! So, keep this order in mind as we solve our problem.

Why is the Order of Operations Important?

The order of operations isn't just some arbitrary rule made up to make math harder. It's a fundamental principle that ensures consistency and clarity in mathematical calculations. Without it, we'd have a chaotic mess of different interpretations and results for the same problem. Imagine trying to build a bridge if engineers used different orders of operations – it wouldn't exactly be a sturdy structure, would it? By adhering to PEMDAS/BODMAS, we create a universal language for math, ensuring that everyone arrives at the same solution.

Think of it like a recipe. If the instructions aren't followed in the correct order, you might end up with a culinary disaster instead of a delicious meal. Similarly, in math, the order of operations provides the recipe for solving equations, guiding us through the steps to reach the correct answer. This systematic approach not only helps us solve complex problems but also develops our logical thinking and problem-solving skills. So, next time you see a math problem, remember PEMDAS/BODMAS – it's your secret weapon to success!

Step 1: Tackling the Parentheses

Okay, the first thing we spot in our expression 8 + (-5) - (-6 * 3) + (7 - 10) are those parentheses. According to PEMDAS/BODMAS, this is where we start our adventure. We've got two sets of parentheses to deal with: (-5) and (7 - 10).

The first one, (-5), is pretty straightforward. It's just a negative number hanging out in parentheses. Sometimes parentheses are used to simply clarify a sign, and that's what's happening here. So, we can essentially just think of it as -5.

Now, let's tackle the second set: (7 - 10). This is a mini-subtraction problem within the larger expression. When we subtract 10 from 7, we get -3. Think of it like starting at 7 on a number line and moving 10 steps to the left – you'll land on -3. So, we can replace (7 - 10) with -3 in our expression.

Why do Parentheses Come First?

Ever wondered why parentheses get the VIP treatment in the order of operations? Well, they're like the bosses of the math world, dictating which operations need to be prioritized. Parentheses act as grouping symbols, telling us to treat the enclosed expression as a single unit before anything else. This is crucial because it allows us to override the default order of operations and control how the problem is solved.

Imagine you have the expression 2 + 3 * 4. Without parentheses, we'd multiply 3 * 4 first (according to PEMDAS/BODMAS) and then add 2, resulting in 14. But if we add parentheses like this: (2 + 3) * 4, we're telling the math world, "Hey, do this addition first!" So, we add 2 and 3 to get 5, and then multiply by 4, giving us 20. See how the parentheses completely change the outcome?

This ability to group and prioritize operations is incredibly powerful. It allows us to create complex expressions and solve them in a logical, controlled manner. Parentheses are like the punctuation marks of math, guiding us through the expression and ensuring we understand the intended order of operations. So, always remember to give those parentheses the respect they deserve – they're the key to unlocking many mathematical mysteries!

Step 2: Multiplication Time!

Alright, we've conquered the parentheses, and our expression is looking a bit cleaner: 8 + (-5) - (-6 * 3) + (-3). Now, if we peek at PEMDAS/BODMAS again, we see that the next operation on our list is multiplication. We've got one multiplication operation lurking in our expression: (-6 * 3).

When we multiply -6 by 3, we get -18. Remember the rules for multiplying integers: a negative times a positive equals a negative. So, we can replace (-6 * 3) with -18. Our expression now transforms into 8 + (-5) - (-18) + (-3).

The Importance of Integer Rules in Multiplication

When dealing with multiplication (and division) in math, it's crucial to remember the rules for working with integers – those positive and negative numbers that add a bit of zest to our calculations. These rules are like the traffic laws of the math world, guiding us to the correct destination. Let's break them down:

• Positive times Positive: A positive number multiplied by a positive number always results in a positive number. Think of it as adding groups of positive things – you'll always end up with something positive.
• Negative times Negative: A negative number multiplied by a negative number also gives us a positive number. This might seem a bit counterintuitive at first, but it's a fundamental rule of math. Imagine subtracting a negative debt – it's like getting money back!
• Positive times Negative (or Negative times Positive): When we multiply a positive number by a negative number (or vice versa), the result is always negative. This is because we're essentially adding groups of negative things, leading to a negative outcome.

These rules are essential for accurately solving mathematical expressions. If we forget them, we might end up with the wrong sign in our answer, leading to a completely different result. So, it's worth taking the time to memorize and understand these rules – they'll be your trusty companions in the world of math!

Step 3: Addition and Subtraction – The Final Stretch

We're on the home stretch now! Our expression is 8 + (-5) - (-18) + (-3). According to PEMDAS/BODMAS, the last operations we need to handle are addition and subtraction. And here's a key point: we tackle these operations from left to right, just like reading a sentence.

First up, we have 8 + (-5). Adding a negative number is the same as subtracting, so this is like saying 8 - 5, which equals 3. Our expression now looks like 3 - (-18) + (-3).

Next, we have 3 - (-18). Subtracting a negative number is the same as adding its positive counterpart. So, this becomes 3 + 18, which equals 21. Our expression simplifies to 21 + (-3).

Finally, we have 21 + (-3). Again, adding a negative number is the same as subtracting, so this is like saying 21 - 3, which equals 18. Woohoo! We've reached the finish line!

Left to Right: Why Order Matters in Addition and Subtraction

You might be wondering, "Why do we need to do addition and subtraction from left to right? Can't we just add and subtract in any order we want?" Well, the answer is a resounding no! The order in which we perform these operations can significantly impact the final result. Let's see why.

Imagine you have the expression 10 - 4 + 2. If we follow the left-to-right rule, we first subtract 4 from 10, getting 6. Then, we add 2 to 6, resulting in 8. But what if we decided to add 4 and 2 first, and then subtract that from 10? We'd get 10 - 6, which equals 4 – a completely different answer!

This difference arises because subtraction and addition are inverse operations, and their order matters. Think of it like walking a path – if you take a few steps forward and then a few steps back, you might not end up in the same place if you did it in reverse. Similarly, in math, performing subtraction and addition in the correct order ensures we accurately account for the changes in value.

The left-to-right rule provides a consistent and unambiguous way to handle addition and subtraction, preventing confusion and ensuring everyone arrives at the same solution. It's like a set of traffic laws for mathematical operations, keeping things flowing smoothly and preventing collisions. So, always remember to follow the left-to-right rule when adding and subtracting – it's the key to mathematical harmony!

Conclusion: The Final Answer

So, after our mathematical journey through parentheses, multiplication, addition, and subtraction, we've arrived at our final answer: 18. We successfully solved the expression 8 + (-5) - (-6 * 3) + (7 - 10) by carefully following the order of operations (PEMDAS/BODMAS) and tackling each step methodically.

Remember, guys, math problems might seem tricky at first glance, but breaking them down into smaller, manageable steps makes them a whole lot easier. And the order of operations is our trusty guide, ensuring we solve them accurately every time. Keep practicing, and you'll become math wizards in no time! Great job today!