Solving Algebraic Expressions A Step-by-Step Guide To (4a-2b)-(a+5b)

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Hey guys! Let's dive into this math problem together. We've got a classic algebraic expression here: (4a-2b)-(a+5b). If you're feeling a bit puzzled, don't worry! We're going to break it down step-by-step so it all makes sense. This type of problem is super common in algebra, and mastering it will definitely help you ace those math tests and impress your teacher. So, let's get started!

What are Algebraic Expressions?

First things first, let's quickly recap what algebraic expressions actually are. In simple terms, they're mathematical phrases that combine numbers, variables (like our 'a' and 'b'), and operation signs (+, -, ×, ÷). The beauty of algebra is that it allows us to represent unknown quantities with variables, which opens up a whole new world of problem-solving possibilities.

In our expression, (4a-2b)-(a+5b), 'a' and 'b' are the variables. The numbers in front of the variables (like 4 and -2) are called coefficients. And the operations we're dealing with here are subtraction and addition. The key to solving these expressions is to simplify them by combining like terms. But before we jump into that, let's understand why this stuff matters.

Algebra isn't just some abstract concept you learn in school and then forget. It's actually incredibly useful in real life! From calculating the cost of groceries to figuring out the trajectory of a rocket, algebra is the backbone of many everyday calculations and advanced scientific applications. Mastering algebraic expressions like (4a-2b)-(a+5b) builds a solid foundation for more complex math and science topics down the road.

Step-by-Step Solution of (4a-2b)-(a+5b)

Okay, let's get down to business and solve this expression! We'll take it one step at a time to make sure everything is crystal clear.

Step 1: Distribute the Negative Sign

This is the crucial first step that many students sometimes overlook. We have a negative sign in front of the parentheses (a+5b), which means we need to distribute that negative sign to each term inside the parentheses. Think of it as multiplying each term inside the parentheses by -1.

So, -(a+5b) becomes -a-5b. Remember, a negative times a positive is a negative!

Now our expression looks like this: 4a-2b-a-5b. See how we've removed the parentheses? That's a big step forward.

Step 2: Identify Like Terms

Next up, we need to identify the like terms in our expression. Like terms are terms that have the same variable raised to the same power. In our case, we have two types of like terms:

  • Terms with 'a': 4a and -a
  • Terms with 'b': -2b and -5b

It's helpful to group these terms together in your mind (or even rewrite the expression with the like terms next to each other) so you don't accidentally combine terms that shouldn't be combined.

Step 3: Combine Like Terms

Now for the fun part: combining those like terms! This simply means adding or subtracting the coefficients of the like terms.

Let's start with the 'a' terms: 4a - a. Remember that if there's no coefficient written in front of a variable, it's understood to be 1. So, -a is the same as -1a. Therefore, 4a - a is the same as 4a - 1a, which equals 3a.

Next, let's combine the 'b' terms: -2b - 5b. This is like adding two negative numbers. Think of it as owing someone $2 and then owing them another $5. You now owe them a total of $7. So, -2b - 5b equals -7b.

Step 4: Write the Simplified Expression

We've done the hard work! Now we just need to put our simplified terms together. We have 3a and -7b, so our final simplified expression is:

3a - 7b

And that's it! We've successfully simplified the expression (4a-2b)-(a+5b) to 3a - 7b. Give yourself a pat on the back!

Common Mistakes to Avoid

Before we wrap up, let's quickly go over some common mistakes that students make when solving problems like this. Being aware of these pitfalls can help you avoid them yourself.

  • Forgetting to Distribute the Negative Sign: This is probably the most common mistake. Remember, that negative sign in front of the parentheses changes the sign of every term inside the parentheses. Don't just change the sign of the first term!
  • Combining Unlike Terms: You can only combine terms that have the same variable raised to the same power. You can't combine 'a' terms with 'b' terms, for example. It's like trying to add apples and oranges – they're just not the same!
  • Sign Errors: Be extra careful with your signs, especially when dealing with negative numbers. A small sign error can throw off your entire answer.
  • Skipping Steps: It can be tempting to try to do everything in your head, but it's much safer to write out each step, especially when you're first learning. This helps you keep track of your work and reduces the chances of making a mistake.

Practice Makes Perfect

The best way to master algebraic expressions is to practice, practice, practice! The more problems you solve, the more comfortable you'll become with the steps involved. Look for practice problems in your textbook or online, and don't be afraid to ask your teacher or a classmate for help if you get stuck.

Here are a few more problems you can try on your own:

  • (2x + 3y) - (x - y)
  • (5p - 2q) + (3p + 4q)
  • (7m + n) - (2m - 3n)

Remember to follow the same steps we used above: distribute the negative sign, identify like terms, combine like terms, and write the simplified expression.

Conclusion

So, there you have it! We've walked through the solution of (4a-2b)-(a+5b) step-by-step, and we've also discussed some common mistakes to avoid and the importance of practice. I hope this guide has been helpful and that you're feeling more confident about tackling algebraic expressions. Keep up the great work, and you'll be an algebra pro in no time! Remember, math can be fun, especially when you understand it. Keep practicing, keep asking questions, and keep exploring the amazing world of mathematics!

If you have any other math questions, feel free to ask! Good luck with your studies, and I hope you nail that assignment! Remember, understanding algebra is like unlocking a superpower – it opens doors to so many other exciting fields of study and real-world applications. So, embrace the challenge, and enjoy the journey of learning!