Simplify: (2a - 3b + 4) - (4a + 2b - 3)

by Jhon Lennon 40 views

Hey guys! Today, we're going to dive into simplifying algebraic expressions. Specifically, we'll tackle the expression (2a - 3b + 4) - (4a + 2b - 3). Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step so you can follow along easily. Our goal is to combine like terms and simplify the expression to its simplest form. So, grab your pencil and paper, and let's get started!

Understanding the Basics

Before we jump into the problem, let's quickly review some basic concepts. Remember, in algebra, like terms are terms that have the same variable raised to the same power. For example, 2a and 4a are like terms because they both have the variable 'a' raised to the power of 1. Similarly, 3b and 2b are like terms because they both have the variable 'b' raised to the power of 1. Constants, like 4 and -3, are also like terms because they don't have any variables.

When simplifying expressions, we can only combine like terms. This means we can add or subtract the coefficients (the numbers in front of the variables) of like terms. For example, 2a + 4a = 6a. We simply add the coefficients (2 and 4) to get 6, and then keep the variable 'a'.

Another important concept to remember is the distributive property. When we have a negative sign in front of parentheses, we need to distribute that negative sign to each term inside the parentheses. For example, -(4a + 2b - 3) becomes -4a - 2b + 3. We're essentially multiplying each term inside the parentheses by -1.

With these basics in mind, we're ready to tackle the expression (2a - 3b + 4) - (4a + 2b - 3).

Step-by-Step Solution

Let's break down the expression (2a - 3b + 4) - (4a + 2b - 3) step-by-step:

Step 1: Distribute the Negative Sign

The first thing we need to do is distribute the negative sign in front of the second set of parentheses. This means we need to change the sign of each term inside the parentheses:

(2a - 3b + 4) - (4a + 2b - 3) becomes 2a - 3b + 4 - 4a - 2b + 3

Notice how the signs of the terms inside the second set of parentheses have changed. 4a became -4a, 2b became -2b, and -3 became +3.

Step 2: Group Like Terms

Now that we've distributed the negative sign, we need to group like terms together. This means we need to rearrange the terms so that the 'a' terms are together, the 'b' terms are together, and the constants are together:

2a - 3b + 4 - 4a - 2b + 3 becomes 2a - 4a - 3b - 2b + 4 + 3

We've simply rearranged the terms so that like terms are next to each other. This makes it easier to combine them in the next step.

Step 3: Combine Like Terms

Now we can combine the like terms. Remember, we can only combine terms that have the same variable raised to the same power. So, we'll combine the 'a' terms, the 'b' terms, and the constants separately:

  • Combine the 'a' terms: 2a - 4a = -2a
  • Combine the 'b' terms: -3b - 2b = -5b
  • Combine the constants: 4 + 3 = 7

Step 4: Write the Simplified Expression

Now that we've combined all the like terms, we can write the simplified expression:

-2a - 5b + 7

That's it! The simplified form of the expression (2a - 3b + 4) - (4a + 2b - 3) is -2a - 5b + 7.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are a few common mistakes that you should try to avoid:

  • Forgetting to Distribute the Negative Sign: This is one of the most common mistakes. Remember to distribute the negative sign to every term inside the parentheses. If you forget to do this, you'll end up with the wrong answer.
  • Combining Unlike Terms: You can only combine like terms. Make sure that the terms you're combining have the same variable raised to the same power. For example, you can't combine 2a and 3b because they have different variables.
  • Incorrectly Adding or Subtracting Coefficients: When combining like terms, make sure you add or subtract the coefficients correctly. Pay attention to the signs of the coefficients. For example, 2a - 4a = -2a, not 2a.
  • Dropping Variables: When combining like terms, make sure you keep the variable. For example, 2a + 4a = 6a, not just 6.

By avoiding these common mistakes, you'll be well on your way to simplifying algebraic expressions like a pro!

Practice Problems

Want to test your skills? Try simplifying these expressions:

  1. (3x + 2y - 1) - (x - y + 4)
  2. (5a - 4b + 2) + (2a + 3b - 5)
  3. (2p + 3q - 7) - (5p - q + 2)

See if you can get the correct answers! The solutions are below:

Solutions

  1. 2x + 3y - 5
  2. 7a - b - 3
  3. -3p + 4q - 9

Conclusion

Simplifying algebraic expressions is a fundamental skill in algebra. By understanding the basics, avoiding common mistakes, and practicing regularly, you can master this skill and excel in your math studies. Remember, the key is to distribute the negative sign correctly, group like terms, and combine them carefully.

So, the final answer of simplifying (2a - 3b + 4) - (4a + 2b - 3) is -2a - 5b + 7. Keep practicing, and you'll become a pro in no time! Keep rocking it!