Solving The Sum: (1/3 X - 8) + (1/6 X - 12)

Hey guys! Today, we are going to dive into a fun little math problem where we need to find the sum of an algebraic expression. Don't worry, it's not as intimidating as it sounds! We'll break it down step by step so it’s super easy to follow. Our mission, should we choose to accept it, is to calculate the sum of the expression: (1/3 x - 8) + (1/6 x - 12). So, grab your thinking caps, and let's get started!

Breaking Down the Expression

Before we jump into solving, let’s take a good look at our expression. We have two parts here: (1/3 x - 8) and (1/6 x - 12). Both of these are algebraic expressions, which simply means they contain variables (like our 'x') and constants (those regular numbers like 8 and 12). To find the sum, we need to combine these two parts together. The main keywords to remember here are sum, algebraic expression, variables, and constants. These are the building blocks of our problem, and understanding them will make the whole process much smoother. Remember, math is like building with LEGOs – each piece has its place, and once you understand how they fit, you can create amazing things!

Identifying Like Terms

The first step in making this addition easier is to identify like terms. Like terms are those that have the same variable raised to the same power. In our expression, we have terms with 'x' and constant terms (numbers without a variable). So, 1/3 x and 1/6 x are like terms because they both have 'x' to the power of 1. Similarly, -8 and -12 are like terms because they are both constants. Identifying these like terms is crucial because we can only add or subtract terms that are alike. Trying to add 1/3 x to -8 would be like trying to fit a square peg in a round hole – it just doesn't work! Keeping this concept of like terms in mind will prevent a lot of common mistakes and make simplifying expressions a breeze.

Combining Like Terms with 'x'

Okay, let's get to the actual math! We'll start by combining the terms with 'x': 1/3 x and 1/6 x. To add these, we need a common denominator. Think of it like adding fractions – you can't add them directly unless they have the same denominator. The least common denominator for 3 and 6 is 6. So, we need to convert 1/3 x to have a denominator of 6. We can do this by multiplying both the numerator and the denominator by 2: (1/3) * (2/2) x = 2/6 x. Now we can easily add it to 1/6 x: 2/6 x + 1/6 x. When adding fractions with the same denominator, you simply add the numerators and keep the denominator the same. So, 2/6 x + 1/6 x = (2+1)/6 x = 3/6 x. We're not quite done yet! We can simplify 3/6 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, 3/6 x simplifies to 1/2 x. Awesome! We've just combined the 'x' terms and simplified the result. Feels good, right?

Combining Constant Terms

Now, let's tackle the constant terms: -8 and -12. This part is a bit simpler because we’re just adding regular numbers. Remember that we're adding a negative number, so it's like subtracting. Think of a number line – if you start at -8 and move 12 units further to the left (in the negative direction), where do you end up? You end up at -20. So, -8 + (-12) = -20. Easy peasy! We’ve now successfully combined our constant terms. This step is crucial because it helps us simplify the expression into its most basic form. Make sure to pay close attention to the signs (positive or negative) when combining constants, as a small mistake here can change the whole answer.

Putting It All Together

We've done the hard work of combining like terms separately. Now, it’s time to put everything together to get our final answer. We found that 1/3 x + 1/6 x = 1/2 x and -8 + (-12) = -20. So, we simply combine these results to form our simplified expression. Our final sum is 1/2 x - 20. And there you have it! We've successfully solved the expression. Remember, the key to these problems is to break them down into smaller, manageable steps. First, identify like terms, then combine them, and finally, put the simplified terms together. This process not only makes the problem easier to solve but also helps you understand the underlying concepts better.

Checking Our Work

It's always a good idea to double-check our work, just to be sure we didn't make any silly mistakes along the way. One way to do this is to plug in a value for 'x' into both the original expression and our simplified expression. If we get the same result in both cases, it's a good indication that we've done everything correctly. Let's try plugging in x = 6. In the original expression, we have:

(1/3 * 6 - 8) + (1/6 * 6 - 12)

Let's simplify this step by step:

(2 - 8) + (1 - 12)

-6 + (-11)

-17

Now, let’s plug x = 6 into our simplified expression, 1/2 x - 20:

(1/2 * 6 - 20)

3 - 20

-17

Ta-da! We got the same result in both cases, which means we can be pretty confident that our answer is correct. Checking your work is a great habit to develop because it catches any errors early on and ensures you’re submitting accurate solutions. Plus, it gives you that extra peace of mind knowing you've nailed the problem.

Common Mistakes to Avoid

When solving expressions like this, there are a few common mistakes that students often make. Let’s go over them so you can avoid these pitfalls. One frequent error is forgetting to find a common denominator when adding fractions. Remember, you can’t add 1/3 x and 1/6 x directly without converting them to fractions with the same denominator. Another common mistake is mixing up the signs when adding constant terms. For example, students might accidentally add -8 + (-12) and get a positive number instead of -20. Always double-check your signs! A third error is not simplifying the final expression completely. In our case, we simplified 3/6 x to 1/2 x. Make sure you always reduce fractions to their simplest form. By being aware of these common mistakes, you can actively work to avoid them and improve your accuracy.

Real-World Applications

You might be wondering, “Okay, this is cool, but when am I ever going to use this in real life?” Well, believe it or not, algebraic expressions are used in many everyday situations. For example, imagine you're planning a road trip. You need to calculate the total cost, which might include gas, food, and accommodation. Each of these costs can be represented using variables, and you might need to combine them to get the total expense. Similarly, in cooking, you might need to adjust recipe quantities based on the number of people you’re serving. This often involves adding or subtracting fractions and whole numbers, just like we did in our expression. Understanding how to work with algebraic expressions is also crucial in fields like finance, engineering, and computer science. So, the skills you’re learning now are laying the foundation for many future applications. Math isn’t just about numbers and equations; it’s a powerful tool for problem-solving in all aspects of life.

Practice Makes Perfect

Like any skill, mastering algebraic expressions takes practice. The more you practice, the more comfortable and confident you’ll become. Try working through similar problems on your own. You can find plenty of examples in textbooks, online resources, or even create your own problems. Challenge yourself by changing the numbers or adding more terms to the expression. The key is to consistently apply the steps we’ve discussed: identify like terms, combine them carefully, and simplify the result. Don’t get discouraged if you make mistakes – they’re a natural part of the learning process. Just review your work, identify where you went wrong, and try again. With enough practice, you’ll be solving these expressions like a pro in no time!

Conclusion

So, guys, we've successfully calculated the sum of the expression (1/3 x - 8) + (1/6 x - 12), and the answer is 1/2 x - 20. We broke down the problem step by step, identified like terms, combined them, and even checked our work. Remember, the key to solving algebraic expressions is to take your time, be organized, and pay attention to detail. Math might seem challenging at times, but with a little practice and the right approach, you can conquer any problem. Keep up the great work, and happy solving!