Simplifying Algebraic Expressions Step-by-Step Solutions

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Hey guys! Ever feel like algebraic expressions are just a jumbled mess of letters and numbers? Don't worry, you're not alone! Simplifying these expressions is a fundamental skill in mathematics, and once you get the hang of it, it's like unlocking a secret code. In this article, we'll break down the process step-by-step, making it super easy to understand. We'll tackle expressions involving distribution, division, and subtraction, ensuring you're a pro at simplifying in no time. So, buckle up and let's dive into the world of algebraic simplification!

Understanding the Basics of Algebraic Expressions

Before we jump into the nitty-gritty of simplifying, let's make sure we're all on the same page regarding what algebraic expressions actually are. At their core, algebraic expressions are combinations of variables, constants, and mathematical operations. Variables are those sneaky letters (like x, y, or a) that represent unknown values. Constants are the numbers that stand alone, like 2, 7, or -5. And the mathematical operations are the usual suspects: addition, subtraction, multiplication, and division.

Now, the key to simplifying algebraic expressions lies in understanding the order of operations and the distributive property. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? It's your trusty guide through the maze of operations. The distributive property, on the other hand, allows us to multiply a term across a sum or difference within parentheses. These two concepts are the cornerstones of simplification, and we'll be using them extensively throughout this article.

Also, it's crucial to remember the importance of combining like terms. Like terms are those that have the same variable raised to the same power. For example, 3x and 5x are like terms because they both have 'x' raised to the power of 1. We can combine them by simply adding or subtracting their coefficients (the numbers in front of the variables). However, 3x and 5x² are not like terms because the powers of 'x' are different. Combining like terms is like tidying up your expression, making it cleaner and easier to work with.

Think of it like this: you can only add apples to apples and oranges to oranges. You can't just lump them all together! Similarly, you can only combine terms that have the exact same variable part. Mastering the identification and combination of like terms is a major step towards simplifying complex algebraic expressions. It's like finding the hidden order within the chaos, making the problem much more manageable.

Problem 1: Simplifying (1/6)(12a + 6b – 18)

Let's kick things off with our first problem: (1/6)(12a + 6b – 18). This expression involves the distributive property, which is our main tool for simplification here. The distributive property states that a(b + c) = ab + ac. In simpler terms, it means we need to multiply the term outside the parentheses (in this case, 1/6) by each term inside the parentheses.

So, let's apply this to our problem. We'll multiply 1/6 by 12a, then by 6b, and finally by -18. This gives us: (1/6) * 12a + (1/6) * 6b + (1/6) * (-18). Now, let's perform these multiplications. Remember that multiplying a fraction by a whole number is like dividing the whole number by the denominator of the fraction.

(1/6) * 12a simplifies to 2a because 12 divided by 6 is 2. Similarly, (1/6) * 6b simplifies to b because 6 divided by 6 is 1. And finally, (1/6) * (-18) simplifies to -3 because -18 divided by 6 is -3. So, our expression now looks like this: 2a + b - 3.

Now, here's the crucial question: can we simplify this any further? Look closely at the terms. Do we have any like terms that can be combined? Nope! We have a term with 'a', a term with 'b', and a constant term. These are all different, so we can't combine them. This means that 2a + b - 3 is the simplest form of our expression. We've successfully used the distributive property and identified like terms (or the lack thereof) to arrive at our final answer. Isn't it satisfying when things come together so neatly?

Problem 2: Simplifying (8x + 12y – 4) ÷ 4

Alright, let's move on to our second challenge: (8x + 12y – 4) ÷ 4. This problem involves division, but don't let that scare you! We can tackle this in a similar way to the previous one, using a slightly different perspective. Remember that dividing by a number is the same as multiplying by its reciprocal. The reciprocal of 4 is 1/4, so we can rewrite our expression as (1/4)(8x + 12y – 4).

Hey, wait a minute! This looks familiar, doesn't it? We've transformed our division problem into a multiplication problem, and we already know how to handle those! We can use the distributive property, just like we did in the first problem. We'll multiply 1/4 by each term inside the parentheses.

So, let's do it! (1/4) * 8x + (1/4) * 12y + (1/4) * (-4). Now, let's perform the multiplications. (1/4) * 8x simplifies to 2x because 8 divided by 4 is 2. (1/4) * 12y simplifies to 3y because 12 divided by 4 is 3. And (1/4) * (-4) simplifies to -1 because -4 divided by 4 is -1. Our expression now looks like this: 2x + 3y - 1.

Time for our key question again: can we simplify this any further? Take a close look. Do we have any like terms lurking around? Nope! We have a term with 'x', a term with 'y', and a constant term. These are all different, so we can't combine them. This means that 2x + 3y - 1 is the simplest form of our expression. We've successfully transformed division into multiplication, applied the distributive property, and identified the absence of like terms to reach our simplified answer. High five!

Problem 3: Simplifying 4x + 5 – (15x – 18) ÷ 3

Okay, guys, let's tackle our final problem, which is a bit more complex: 4x + 5 – (15x – 18) ÷ 3. This one throws in subtraction and division, so we need to be extra careful with our order of operations. Remember PEMDAS? Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

First, we need to deal with the division inside the parentheses. We have (15x – 18) ÷ 3. Just like in the previous problem, we can rewrite this as multiplication: (1/3)(15x – 18). Now we can apply the distributive property. (1/3) * 15x + (1/3) * (-18) simplifies to 5x - 6. So, our expression now looks like this: 4x + 5 – (5x - 6).

Now comes the tricky part: the subtraction of the entire expression in parentheses. Remember that subtracting a group of terms is the same as adding the negative of each term. So, we can rewrite the expression as 4x + 5 - 5x + 6. Notice how the sign of each term inside the parentheses has changed. This is a crucial step, and a common place to make mistakes, so pay close attention!

Now that we've handled the subtraction, it's time to combine like terms. We have two terms with 'x': 4x and -5x. Combining them gives us -1x, which we can simply write as -x. We also have two constant terms: 5 and 6. Combining them gives us 11. So, our expression now looks like this: -x + 11.

Can we simplify this any further? Nope! We have a term with 'x' and a constant term. These are different, so we can't combine them. This means that -x + 11 is the simplest form of our expression. We've successfully navigated the order of operations, handled subtraction carefully, and combined like terms to arrive at our final simplified answer. You're crushing it!

Key Takeaways for Simplifying Algebraic Expressions

Alright, guys, we've covered a lot of ground in this article! Let's recap the key takeaways to solidify your understanding of simplifying algebraic expressions:

  • Master the Order of Operations (PEMDAS): This is your guiding principle. Always tackle parentheses first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Think of it as the roadmap for your simplification journey.
  • Embrace the Distributive Property: This is your secret weapon for dealing with parentheses. Remember to multiply the term outside the parentheses by every term inside. It's like sharing the love (or the multiplication) with everyone in the group.
  • Become a Like Term Detective: Identifying and combining like terms is crucial for simplifying. Remember, like terms have the same variable raised to the same power. You can only add apples to apples, not apples to oranges!
  • Pay Attention to Subtraction: Subtracting a group of terms is like distributing a negative sign. Remember to change the sign of every term inside the parentheses. This is a common pitfall, so stay sharp!
  • Practice, Practice, Practice: The more you practice, the more comfortable you'll become with simplifying algebraic expressions. It's like learning a new language; the more you use it, the more fluent you'll become.

Simplifying algebraic expressions might seem daunting at first, but with a solid understanding of the basics and consistent practice, you'll be simplifying like a pro in no time. Keep these key takeaways in mind, and you'll be well on your way to conquering any algebraic challenge that comes your way. You've got this!

Practice Problems for You to Solve

To really solidify your understanding, here are a few practice problems for you to try on your own. Remember to apply the principles we've discussed in this article, and don't be afraid to make mistakes! Mistakes are learning opportunities in disguise.

  1. Simplify: (1/4)(8x - 12y + 20)
  2. Simplify: (10a + 15b - 5) ÷ 5
  3. Simplify: 2x + 7 – (9x – 12) ÷ 3

Good luck, and happy simplifying! You can do it!

Conclusion

We've journeyed through the world of simplifying algebraic expressions, tackling distribution, division, subtraction, and the crucial task of combining like terms. Remember, algebraic simplification isn't just about following rules; it's about understanding the underlying principles and applying them strategically. By mastering these techniques, you're not just solving problems; you're building a strong foundation for more advanced mathematical concepts. So, keep practicing, keep exploring, and keep simplifying! The world of algebra awaits your mastery.