Combining Like Terms: A Step-by-Step Guide

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Hey guys! Ever stumbled upon an expression that looks like a jumbled mess of numbers and variables? Don't worry, we've all been there! The secret to untangling these mathematical knots lies in combining like terms. In this guide, we'll break down the process step-by-step, using the example expression 20(−1.5r+0.75)20(-1.5 r+0.75) as our trusty sidekick. Get ready to transform that complicated expression into a sleek, simplified version! Understanding how to combine like terms is a foundational skill in algebra, paving the way for tackling more complex equations and problems. It's not just about getting the right answer; it's about understanding the structure of mathematical expressions and how they work. So, let's dive in and make math a little less intimidating, one term at a time!

What are Like Terms Anyway?

Before we jump into the expression itself, let's make sure we're crystal clear on what like terms actually are. Think of them as mathematical twins – they have the same variable raised to the same power. For example, 3x and -5x are like terms because they both have the variable x raised to the power of 1 (which is usually not explicitly written). On the other hand, 3x and 3x² are not like terms because, while they both have x, the powers are different. Similarly, 2y and 2z are not like terms because they involve different variables.

Constants, which are just numbers without any variables, are also considered like terms. So, 5 and -7 are like terms. The beauty of identifying like terms is that we can combine them through addition or subtraction. This is where the magic of simplification happens!

Remember, the key is to look for terms that have the exact same variable part. The coefficient (the number in front of the variable) can be different, but the variable and its exponent must match for terms to be considered like terms. Mastering this concept is the first step toward simplifying algebraic expressions like a pro. Without a solid grasp of like terms, the simplification process can become confusing and error-prone. So, take a moment to really let this sink in, and you'll be well-equipped to tackle the challenges ahead. Think of it as building a strong foundation for your mathematical house – the stronger the foundation, the more confidently you can build upon it!

Our Mission: Simplifying 20(−1.5r+0.75)20(-1.5 r+0.75)

Now that we're experts on like terms, let's get back to our mission: simplifying the expression 20(−1.5r+0.75)20(-1.5 r+0.75). This expression might look a bit intimidating at first glance, but don't worry, we're going to break it down into manageable steps. The first thing we need to do is tackle those parentheses. Whenever you see parentheses in an expression, it's a big hint that the distributive property is about to come into play. The distributive property is our secret weapon for multiplying a single term by an expression inside parentheses. It basically says that a(b + c) = ab + ac. In plain English, we multiply the term outside the parentheses by each term inside the parentheses.

So, in our case, we need to distribute the 20 across both terms inside the parentheses: -1.5r and 0.75. Let's do it! First, we multiply 20 by -1.5r: 20 * (-1.5r) = -30r. Next, we multiply 20 by 0.75: 20 * 0.75 = 15. Now, we can rewrite our expression as -30r + 15. Ta-da! We've successfully eliminated the parentheses and made the expression look much simpler. This step is crucial because it sets the stage for combining any like terms that might be lurking in the expression. Without distributing first, we wouldn't be able to properly identify and combine those terms. Think of the distributive property as the key that unlocks the potential for simplification. It's a fundamental tool in algebra, and mastering it will make your mathematical life much easier. So, let's embrace the power of distribution and move on to the next step in our simplification journey!

Unleashing the Distributive Property

Let's delve deeper into the magic of the distributive property. This property is your best friend when dealing with expressions containing parentheses, as it allows you to multiply a single term by multiple terms within the parentheses. This is particularly useful for eliminating parentheses and making expressions easier to work with. Remember the golden rule: a(b + c) = ab + ac. This means we multiply the term outside the parentheses (a) by each term inside (b and c), and then add the results together.

In our expression, 20(−1.5r+0.75)20(-1.5 r+0.75), the term outside the parentheses is 20, and the terms inside are -1.5r and 0.75. Let's break down the distribution step by step:

  1. Multiply 20 by -1.5r: 20 * (-1.5r) = -30r
  2. Multiply 20 by 0.75: 20 * 0.75 = 15

Now, we combine these results to get our new expression: -30r + 15. Notice how the parentheses have vanished! The distributive property has worked its magic, transforming the original expression into a more manageable form. This step is essential because it allows us to identify and combine like terms in the next stage of simplification.

The distributive property is not just a mathematical trick; it's a fundamental concept that underlies many algebraic manipulations. It's like the secret sauce that makes complex expressions taste a whole lot better (or, in this case, simpler!). Understanding and applying the distributive property correctly is crucial for success in algebra and beyond. So, make sure you've got this one down pat, and you'll be well on your way to mastering mathematical expressions.

Spotting and Combining Like Terms

After applying the distributive property, our expression has transformed into -30r + 15. Now comes the exciting part: spotting and combining like terms! Remember, like terms are terms that have the same variable raised to the same power. In our simplified expression, we need to carefully examine each term and see if any twins are hiding in plain sight.

Looking at -30r + 15, we have two terms: -30r and 15. The first term, -30r, has the variable r raised to the power of 1. The second term, 15, is a constant – it's just a number without any variables. Are there any other terms in the expression that have the variable r raised to the power of 1? Nope! Are there any other constant terms? Again, nope! In this particular expression, we only have one term with the variable r and one constant term. This means there are no like terms to combine!

Sometimes, you'll encounter expressions with multiple terms that can be combined. For example, if our expression was -30r + 15 + 10r - 5, we would have some like terms to play with. We have two terms with the variable r (-30r and 10r) and two constant terms (15 and -5). In such cases, we would combine the coefficients (the numbers in front of the variables) of the like terms. So, -30r + 10r would become -20r, and 15 - 5 would become 10. Our simplified expression would then be -20r + 10. But in our current expression, -30r + 15, there's no combining to be done. It's already in its simplest form!

Recognizing when terms are "like" is a crucial skill in algebra. It's like being a detective, searching for matching pieces in a puzzle. The more you practice, the better you'll become at spotting like terms and combining them with ease. This skill will not only help you simplify expressions but also solve equations and tackle more advanced mathematical concepts. So, keep your eyes peeled for those mathematical twins, and you'll be a simplification superstar in no time!

The Grand Finale: Our Simplified Expression

After our adventure through the distributive property and the land of like terms, we've reached our destination! Our original expression, 20(−1.5r+0.75)20(-1.5 r+0.75), has been transformed into its simplest, most elegant form. Remember, we started by distributing the 20 across the terms inside the parentheses, which gave us -30r + 15. Then, we carefully examined the expression for like terms, but alas, there were none to combine.

This means that -30r + 15 is our final answer! It's the equivalent expression to the original, but it's much easier to work with. We've successfully simplified the expression by eliminating the parentheses and combining any like terms (or, in this case, recognizing that there weren't any to combine). This process of simplification is a fundamental skill in algebra and is used extensively in solving equations, graphing functions, and tackling more complex mathematical problems.

The journey of simplifying expressions might seem a bit like a puzzle at first, but with practice, it becomes second nature. Each step – applying the distributive property, identifying like terms, and combining them – builds upon the previous one, leading you to the simplified form. And remember, the simplified form is not just a shorter version of the expression; it's a clearer, more understandable representation of the same mathematical idea. It's like taking a cluttered room and organizing it so that everything is in its place. The room (or the expression) is still the same, but it's much easier to navigate and appreciate.

So, congratulations! You've successfully navigated the world of combining like terms and simplified the expression 20(−1.5r+0.75)20(-1.5 r+0.75). You're now one step closer to mastering the art of algebra. Keep practicing, keep exploring, and keep simplifying! The world of mathematics is full of exciting challenges, and you've got the skills to tackle them head-on.

Key Takeaways and Practice

Alright, guys, let's recap what we've learned in this awesome journey of simplifying expressions! Combining like terms is a fundamental skill in algebra, and understanding the process is crucial for success in more advanced math topics. We started with the expression 20(−1.5r+0.75)20(-1.5 r+0.75) and transformed it into its simplified form, -30r + 15. Let's break down the key steps:

  1. Distributive Property: This is your go-to tool for eliminating parentheses. Remember the golden rule: a(b + c) = ab + ac. Multiply the term outside the parentheses by each term inside.
  2. Identify Like Terms: Look for terms that have the same variable raised to the same power. Constants are also considered like terms.
  3. Combine Like Terms: Add or subtract the coefficients (the numbers in front of the variables) of like terms. Remember, you can only combine terms that are alike!

In our example, we distributed the 20 across the terms inside the parentheses, resulting in -30r + 15. Then, we carefully looked for like terms, but since we had only one term with the variable r and one constant term, there was nothing to combine. So, -30r + 15 became our final, simplified expression.

Now, the real magic happens with practice! To solidify your understanding, try simplifying these expressions:

  • 3(2x + 5) - x
  • -2(4y - 1) + 6y
  • 5z + 2(z - 3) + 7

Work through each expression step-by-step, applying the distributive property and combining like terms. Don't be afraid to make mistakes – that's how we learn! The more you practice, the more comfortable and confident you'll become with simplifying expressions. And remember, mastering these foundational skills will set you up for success in all your future mathematical endeavors.

So, grab a pencil and paper, and let's get simplifying! You've got this!