Simplifying 3(4x-5y+6) + 2(3x-5y-6) A Step-by-Step Guide

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Hey guys! Algebra can seem tricky, especially when you're faced with expressions that look like a jumbled mess of numbers, variables, and parentheses. But don't worry, we're going to break down one of these expressions step-by-step so you can see how easy it can actually be. We will focus on simplifying the algebraic expression 3(4x-5y+6) + 2(3x-5y-6). Let's dive in and make algebra a breeze!

Understanding the Basics of Algebraic Expressions

Before we jump into simplifying the expression, let's make sure we're all on the same page with the basics. Algebraic expressions are combinations of variables (like x and y), constants (numbers), and operations (like addition, subtraction, multiplication, and division). The key to simplifying these expressions is to follow the order of operations (PEMDAS/BODMAS) and combine like terms.

  • Variables: These are letters that represent unknown values. In our expression, x and y are the variables.
  • Constants: These are just numbers – they don't change. In our expression, we have constants like 6 and -6.
  • Coefficients: These are the numbers that multiply the variables. For example, in the term 4x, 4 is the coefficient.
  • Like Terms: These are terms that have the same variable raised to the same power. For instance, 4x and 3x are like terms, and -5y and -5y are like terms. Constants are also like terms.

Why is understanding these basics so important? Because simplifying algebraic expressions is all about identifying and combining these different components in the correct way. Think of it like sorting your laundry – you group the socks together, the shirts together, and so on. In algebra, we group the like terms together to make the expression simpler and easier to work with.

To truly master simplifying algebraic expressions, it's crucial to understand the distributive property. This property is the secret weapon for getting rid of those parentheses. The distributive property states that a(b + c) = ab + ac. Basically, you multiply the term outside the parentheses by each term inside the parentheses. This is a fundamental concept that we'll use extensively in simplifying our expression.

Another essential skill is the ability to combine like terms. Like terms, as we discussed earlier, are terms that have the same variable raised to the same power. When combining like terms, you simply add or subtract their coefficients while keeping the variable part the same. For example, 3x + 2x = 5x. It's like saying you have 3 apples and you get 2 more apples – now you have 5 apples!

Finally, 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). This order ensures that we perform operations in the correct sequence, leading to the correct simplified expression. Think of it as a recipe – you need to follow the steps in the right order to bake a delicious cake!

So, with these basics in mind, we're well-prepared to tackle the expression 3(4x-5y+6) + 2(3x-5y-6). Let's move on to the next step and see how the distributive property can help us simplify this expression.

Step 1: Applying the Distributive Property

The first step in simplifying our algebraic expression, 3(4x-5y+6) + 2(3x-5y-6), involves using the distributive property. Remember, this property allows us to multiply the term outside the parentheses by each term inside. It's like sharing the love (or the multiplication!) with everyone in the group.

Let's start with the first part of the expression: 3(4x - 5y + 6). We need to distribute the 3 to each term inside the parentheses. This means we'll multiply 3 by 4x, 3 by -5y, and 3 by 6. Let's break it down:

  • 3 * 4x = 12x
  • 3 * -5y = -15y
  • 3 * 6 = 18

So, 3(4x - 5y + 6) simplifies to 12x - 15y + 18. See? It's like magic! The parentheses have disappeared, and we have a more manageable expression.

Now, let's move on to the second part of the expression: 2(3x - 5y - 6). We'll do the same thing – distribute the 2 to each term inside the parentheses:

  • 2 * 3x = 6x
  • 2 * -5y = -10y
  • 2 * -6 = -12

So, 2(3x - 5y - 6) simplifies to 6x - 10y - 12. We've successfully distributed the 2 and eliminated the parentheses in this part as well.

Now that we've applied the distributive property to both parts of the expression, we have a new, expanded expression: 12x - 15y + 18 + 6x - 10y - 12. It looks a bit long, but don't worry! The next step is all about bringing together the like terms, which will make things much simpler.

The distributive property is a fundamental tool in algebra, and mastering it is crucial for simplifying expressions. It's like having a superpower that allows you to break down complex expressions into smaller, more manageable pieces. By distributing the term outside the parentheses, we've essentially cleared the first hurdle in simplifying our expression. Now, we're ready to move on to the next step: combining like terms. This is where we'll group together the terms that have the same variable and the constants, making our expression even simpler and more elegant. So, let's keep going and see how we can further simplify our expression by combining like terms!

Step 2: Combining Like Terms

Alright, guys, we've made great progress! We've used the distributive property to expand our expression, and now we have: 12x - 15y + 18 + 6x - 10y - 12. The next step is to combine like terms. Remember, like terms are terms that have the same variable raised to the same power, or they are constants. It's like finding all the matching socks in your drawer and putting them together.

Let's start by identifying the like terms in our expression. We have:

  • x terms: 12x and 6x
  • y terms: -15y and -10y
  • Constants: 18 and -12

Now that we've identified the like terms, let's combine them. To combine like terms, we simply add or subtract their coefficients while keeping the variable part the same. It's like adding apples to apples and oranges to oranges – you don't mix them up!

First, let's combine the x terms: 12x + 6x. We add the coefficients (12 and 6) and keep the variable x. So, 12x + 6x = 18x. We've successfully combined our x terms!

Next, let's combine the y terms: -15y - 10y. We add the coefficients (-15 and -10) and keep the variable y. So, -15y - 10y = -25y. Great! We've combined the y terms as well.

Finally, let's combine the constants: 18 - 12. This is a simple subtraction: 18 - 12 = 6. We've combined the constants too!

Now, let's put it all together. We have 18x from combining the x terms, -25y from combining the y terms, and 6 from combining the constants. So, our simplified expression is 18x - 25y + 6. Ta-da! We've successfully combined like terms and simplified our expression.

Combining like terms is a fundamental skill in algebra, and it's like the glue that holds the simplification process together. It allows us to take a complex expression and reduce it to its simplest form. By identifying and combining like terms, we've made our expression much easier to understand and work with. This skill is not only useful for simplifying algebraic expressions but also for solving equations and tackling more advanced algebraic concepts. So, mastering the art of combining like terms is definitely worth the effort!

Step 3: Writing the Simplified Expression

Okay, we've done the heavy lifting! We've applied the distributive property, combined like terms, and now we're at the final step: writing the simplified expression. This is where we present our hard work in a clear and concise way. We've taken the original expression, 3(4x-5y+6) + 2(3x-5y-6), and transformed it into something much simpler. It's like taking a messy room and organizing it into a neat and tidy space.

After applying the distributive property and combining like terms, we arrived at the expression: 18x - 25y + 6. This is our simplified expression. It's the most concise form of the original expression, and it's much easier to work with. We've essentially condensed all the information from the original expression into a more manageable format.

The simplified expression, 18x - 25y + 6, tells us a lot about the relationship between the variables x and y. It shows us how the expression changes as x and y change. It's like having a map that guides us through the algebraic landscape. This simplified form is not only easier to understand but also makes it easier to solve equations, graph functions, and perform other algebraic operations.

Writing the simplified expression is the final step in the simplification process, but it's also a crucial step. It's like putting the finishing touches on a masterpiece. A well-simplified expression is clear, concise, and easy to understand. It's a testament to the power of algebraic manipulation and our ability to transform complex expressions into simpler forms.

So, there you have it! We've successfully simplified the algebraic expression 3(4x-5y+6) + 2(3x-5y-6) into 18x - 25y + 6. We've gone from a jumble of terms and parentheses to a neat and tidy expression that's much easier to work with. This process highlights the power of algebra to simplify complex problems and reveal the underlying relationships between variables and constants.

Conclusion: Mastering Algebraic Simplification

Guys, we did it! We've successfully simplified the algebraic expression 3(4x-5y+6) + 2(3x-5y-6) step-by-step. We started with a seemingly complex expression and, by applying the distributive property and combining like terms, we arrived at the simplified form: 18x - 25y + 6. This journey highlights the beauty and power of algebra in making complex problems more manageable.

Simplifying algebraic expressions is a fundamental skill in mathematics, and it's essential for success in higher-level math courses. It's like learning the alphabet before you can read a book – you need to master the basics before you can tackle more complex concepts. The ability to simplify expressions allows you to solve equations, graph functions, and understand mathematical relationships more easily.

Throughout this guide, we've emphasized the importance of understanding the underlying principles of algebraic simplification. We've broken down the process into clear, manageable steps, and we've explained the reasoning behind each step. By understanding the why behind the how, you'll be better equipped to tackle any algebraic expression that comes your way.

Remember, the key to mastering algebraic simplification is practice, practice, practice! The more you practice, the more comfortable you'll become with the process, and the faster and more accurately you'll be able to simplify expressions. It's like learning a musical instrument – the more you play, the better you become.

So, don't be afraid to tackle those algebraic expressions head-on. With the skills and knowledge you've gained from this guide, you're well on your way to becoming an algebraic simplification master! Keep practicing, keep exploring, and keep simplifying! Algebra can be challenging, but it's also incredibly rewarding when you unlock its secrets. And remember, we're here to help you every step of the way. Happy simplifying!