Solving (2x * 3 + 3x * 2 + 17x * 30) / X * 2

by Jhon Lennon 45 views

Hey guys! Ever stumbled upon an algebraic expression that looks like a confusing mess of numbers and variables? Well, today we're going to break down one of those expressions step by step. Let's dive into how to solve this: (2x * 3 + 3x * 2 + 17x * 30) / x * 2. This might seem daunting at first, but trust me, we'll make it super clear and easy to understand. So, grab your favorite beverage, maybe a notepad, and let's get started!

Understanding the Expression

Before we start crunching numbers, let's quickly understand what we're looking at. The expression (2x * 3 + 3x * 2 + 17x * 30) / x * 2 involves several terms with the variable 'x,' some multiplication, addition, and division. Our goal is to simplify this expression as much as possible. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Keeping this in mind will help us avoid common mistakes. Now, let's break it down piece by piece.

Step 1: Simplify Inside the Parentheses

First, we'll focus on simplifying the terms inside the parentheses. We have three terms to deal with: 2x * 3, 3x * 2, and 17x * 30. Let's simplify each of these individually:

  • 2x * 3 = 6x
  • 3x * 2 = 6x
  • 17x * 30 = 510x

So, now our expression inside the parentheses looks like this: 6x + 6x + 510x. We can further simplify this by adding these terms together. Adding these like terms is a breeze. Think of it like combining similar objects – six apples plus six apples plus 510 apples equals a whole lot of apples! In our case, it's:

6x + 6x + 510x = 522x

Now our entire expression looks much simpler:

(522x) / x * 2

Step 2: Division

Next up, we need to tackle the division. We have 522x divided by x. When we divide 522x by x, the 'x' in the numerator and the 'x' in the denominator cancel each other out. It’s like saying, "For every 'x' we have, we're dividing it away," leaving us with just the coefficient.

522x / x = 522

So now our expression looks even cleaner:

522 * 2

Step 3: Multiplication

Finally, we have a simple multiplication problem. We need to multiply 522 by 2. This is straightforward:

522 * 2 = 1044

So, the simplified answer to our original expression is 1044. Who knew something that looked so complicated could boil down to such a simple number? I hope you're feeling confident about simplifying expressions now!

Practical Examples and Why This Matters

Okay, so we've solved the expression. Great! But why does this even matter? Let's look at some practical examples where simplifying expressions like this can be super useful.

Example 1: Calculating Costs

Imagine you're running a small business selling handmade crafts. You sell two types of items: A and B. The cost of materials for item A is $2 per unit, and you sell 'x' units. The cost of materials for item B is $3 per unit, and you also sell 'x' units. Additionally, you have a fixed shipping cost of $17 per order, and customers order 30 items at a time. Your total cost can be expressed as:

Total Cost = (2 * x + 3 * x + 17 * 30) / x

Simplifying this expression allows you to quickly calculate your average cost per item, which is crucial for pricing your products competitively and ensuring you make a profit. Understanding how to break down this expression helps you manage your business finances more effectively.

Example 2: Engineering and Design

In engineering, you might encounter similar expressions when calculating loads, stresses, or other parameters in a structure. For instance, let's say you're designing a bridge, and you need to calculate the total force acting on a support beam. The force might be represented by an expression like:

Force = (2x * 3 + 3x * 2 + 17x * 30) / x

Here, 'x' could represent a variable related to the weight distribution or some other factor. Simplifying this expression allows you to quickly determine the total force and ensure the bridge is safe and stable.

Example 3: Computer Programming

In programming, you often need to manipulate variables and expressions to perform calculations. Suppose you're writing a program to calculate the total processing time for a series of tasks. The processing time might be expressed as:

Time = (2x * 3 + 3x * 2 + 17x * 30) / x

Here, 'x' could represent the number of data points or some other parameter. Simplifying this expression allows you to optimize your code and reduce processing time, making your program run more efficiently.

Real-World Applications

These examples highlight why understanding algebraic simplification is a valuable skill in various fields. Whether you're managing finances, designing structures, or writing code, being able to quickly and accurately simplify expressions can save you time, reduce errors, and improve your decision-making. So, the next time you see a complicated expression, don't be intimidated. Remember the steps we've covered, and you'll be able to tackle it with confidence!

Common Mistakes to Avoid

Even with a clear understanding of the steps, it's easy to make mistakes. Here are some common pitfalls to watch out for when simplifying expressions like (2x * 3 + 3x * 2 + 17x * 30) / x * 2:

  1. Forgetting the Order of Operations: This is a classic mistake. Always remember PEMDAS/BODMAS. Make sure you perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). For example, if you add before multiplying, you'll end up with the wrong answer.

  2. Incorrectly Combining Like Terms: When adding or subtracting terms, make sure you're only combining like terms. For instance, you can add 6x and 6x because they both have the variable 'x,' but you can't add 6x and 510 because 510 doesn't have an 'x.' Mixing these up can lead to incorrect simplifications.

  3. Dividing Incorrectly: When dividing terms with variables, be careful to cancel out the variables correctly. Remember that x / x = 1, not 0. Forgetting this can lead to errors in your simplification.

  4. Arithmetic Errors: Simple arithmetic mistakes can throw off your entire calculation. Double-check your multiplication, division, addition, and subtraction to ensure accuracy. It's easy to make a small mistake, especially when dealing with larger numbers, so take your time and be meticulous.

  5. Ignoring the Distributive Property: While not directly applicable in this specific expression, the distributive property is crucial in many algebraic simplifications. If you have an expression like a(b + c), you need to distribute the 'a' to both 'b' and 'c,' resulting in ab + ac. Forgetting this property can lead to incorrect simplifications in more complex expressions.

Tips for Avoiding Mistakes

  • Write Clearly: Make sure your handwriting is legible, and clearly write down each step. This helps you avoid misreading your own work.
  • Double-Check Your Work: After each step, take a moment to review what you've done. Catching errors early can save you a lot of time and frustration.
  • Use a Calculator: For complex arithmetic, don't hesitate to use a calculator. This can help you avoid simple calculation errors.
  • Practice Regularly: The more you practice simplifying expressions, the better you'll become at spotting potential mistakes and avoiding them.
  • Ask for Help: If you're unsure about a particular step, don't be afraid to ask for help from a teacher, tutor, or friend. Getting a second opinion can often clarify things and help you avoid errors.

By being aware of these common mistakes and following these tips, you can improve your accuracy and confidence when simplifying algebraic expressions. Keep practicing, and you'll become a pro in no time!

Conclusion

Alright, guys, that wraps up our deep dive into simplifying the expression (2x * 3 + 3x * 2 + 17x * 30) / x * 2. We've seen how breaking down the expression step by step, remembering the order of operations, and avoiding common mistakes can make even the most daunting problems manageable. From simplifying inside the parentheses to handling division and multiplication, each step is crucial to arriving at the correct answer: 1044.

But more than just getting the right answer, we've explored why these skills are so important. Whether it's calculating costs for a small business, designing safe structures in engineering, or optimizing code in computer programming, the ability to simplify algebraic expressions is a valuable asset in many fields. And by being aware of the common pitfalls and following our tips for avoiding mistakes, you can approach these problems with confidence and accuracy.

So, keep practicing, stay curious, and don't be afraid to tackle those complex expressions. You've got this! And remember, every problem you solve is a step closer to mastering the art of algebra. Happy simplifying!