Solve $4(2+3)-2(6-8)+3(7-9)=$ Step-by-Step

Hey math whizzes and curious minds! Today, we're diving into a fun little algebraic expression that's going to test our order of operations skills. You know, the PEMDAS/BODMAS rules we all learned in school? Well, they're about to get a workout! We'll be tackling the expression $4(2+3)-2(6-8)+3(7-9)=$. Don't worry if math gives you the jitters; we're going to break it down step-by-step, making it super clear and easy to follow. So, grab your calculators (or just your brilliant brains!), and let's get started on solving this equation like pros. We'll ensure you understand every single part, from the parentheses to the final calculation. This isn't just about getting the right answer; it's about understanding how we get there, which is the real superpower in mathematics. Ready to conquer this challenge? Let's go!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even touch our specific expression, let's have a quick refresher on the order of operations. You've probably heard of PEMDAS or BODMAS. These acronyms are our guiding stars when solving any mathematical expression involving different types of operations. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS is similar, standing for Brackets, Orders (powers and square roots, etc.), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). The key takeaway here, guys, is that we must follow this sequence. Trying to do things out of order is like trying to build a house starting with the roof – it just doesn't work! For our problem, $4(2+3)-2(6-8)+3(7-9)=$, the parentheses are our first priority. Inside each set of parentheses, we'll perform the operations first. Remember, multiplication and division have equal priority and are done from left to right, as are addition and subtraction.

Step 1: Solving Operations Inside Parentheses

Alright team, let's get down to business with our first step: solving the operations inside the parentheses. This is where PEMDAS/BODMAS tells us to start. We have three sets of parentheses in our expression: $(2+3)$, $(6-8)$, and $(7-9)$. We'll tackle each one individually.

First, let's look at $(2+3)$. This is a straightforward addition. $2 + 3 = 5$. Easy peasy!

Next, we have $(6-8)$. This involves subtraction. When you subtract a larger number from a smaller number, the result is negative. So, $6 - 8 = -2$. It's important to keep that negative sign in mind!

Finally, we look at $(7-9)$. Similar to the previous one, this is another subtraction where we're subtracting a larger number from a smaller one. $7 - 9 = -2$. Again, remember that negative result.

So, after completing this first crucial step, our original expression $4(2+3)-2(6-8)+3(7-9)=$ now transforms into $4(5)-2(-2)+3(-2)=$. See? We're already making great progress by just clearing out those parentheses. Keep that momentum going!

Step 2: Performing Multiplication

Now that we've conquered the parentheses, our next mission is to handle the multiplication. According to PEMDAS/BODMAS, multiplication and division come next, and we perform them from left to right. In our transformed expression, $4(5)-2(-2)+3(-2)=$, we have three multiplication operations to perform. Remember, when a number is directly next to a parenthesis, it means multiplication. Also, a crucial point here is dealing with the signs: multiplying a positive by a negative results in a negative, and multiplying two negatives results in a positive.

Let's start from the left. We have $4(5)$. This means $4 imes 5$, which equals $20$. Pretty simple!

Moving on, we encounter $-2(-2)$. This is where the sign rule is super important. We are multiplying a negative number ($-2$) by another negative number ($-2$). A negative times a negative is a positive. So, $-2 imes -2 = +4$. This is a common spot where mistakes can happen, so always double-check your signs, guys!

Our last multiplication is $3(-2)$. Here, we're multiplying a positive number ($3$) by a negative number ($-2$). A positive times a negative is always negative. So, $3 imes -2 = -6$.

After performing all the multiplications, our expression $4(5)-2(-2)+3(-2)=$ now becomes $20 + 4 - 6=$. We've successfully dealt with all the multiplications. Look at how much simpler the expression looks now! We're just two steps away from our final answer.

Step 3: Performing Addition and Subtraction

We're in the home stretch, folks! The final step in our mathematical journey is to handle the addition and subtraction. PEMDAS/BODMAS tells us to do these from left to right. Our current expression is $20 + 4 - 6=$. This is where we combine our numbers to find the ultimate solution.

We start from the leftmost operation, which is addition: $20 + 4$. This equals $24$.

Now, we take that result and perform the next operation, which is subtraction: $24 - 6$. Calculating this gives us $18$.

And there you have it! The final answer to the expression $4(2+3)-2(6-8)+3(7-9)=$ is $18$. We successfully navigated through the parentheses, multiplication, and finally the addition and subtraction, strictly following the order of operations. It's all about patience and precision, right? Remember these steps the next time you encounter a similar problem. Keep practicing, and you'll become a math ninja in no time!

Conclusion: Mastering Mathematical Expressions

So, there you have it, math enthusiasts! We've successfully demystified the expression $4(2+3)-2(6-8)+3(7-9)=$ by breaking it down using the fundamental rules of mathematics – the order of operations (PEMDAS/BODMAS). It's crucial to remember that mastering these kinds of problems isn't just about memorizing steps; it's about building a strong foundation in logical thinking and problem-solving. Each step, from simplifying inside the parentheses to performing multiplications and finally additions and subtractions, plays a vital role in reaching the correct answer. The initial expression might look intimidating with its combination of numbers and operations, but by applying the rules systematically, we transform it into a manageable sequence of calculations. We saw how crucial it is to handle the signs correctly, especially during multiplication, where a negative multiplied by a negative yields a positive, a common pitfall for many. The left-to-right rule for operations of the same priority, like multiplication/division and addition/subtraction, ensures consistency and accuracy. Ultimately, the answer we arrived at, 18, is a testament to the power of following a structured approach. Keep practicing these types of problems, and you'll find that your confidence and speed will increase significantly. Mathematics is a journey of continuous learning and discovery, and every solved problem is a step forward. So, keep those brains engaged, keep questioning, and keep solving!