Simplifying Algebraic Expressions: Combining Like Terms

In the realm of mathematics, simplifying expressions is a fundamental skill. Today, we're diving into how to combine like terms in an algebraic expression. Combining like terms makes expressions easier to understand and work with. Let's break down the process step-by-step and apply it to the expression: $42 c^2+27 c^2-4 c^2-15 c^2-43 c^2$.

Understanding Like Terms

Before we jump into combining anything, it's super important to know what "like terms" actually are. Like terms are terms that have the same variable raised to the same power. The coefficients (the numbers in front of the variables) can be different, but the variable part must be identical. For example, $3x^2$ and $-5x^2$ are like terms because they both have the variable $x$ raised to the power of 2. On the other hand, $3x^2$ and $3x$ are not like terms because, even though they have the same variable $x$, the powers are different (2 versus 1).

Why do we care about like terms? Because we can only add or subtract terms that are alike. It's like saying you can only add apples to apples, not apples to oranges. In algebraic terms, you can combine $x^2$ terms with other $x^2$ terms, but you can't combine them with $x$ terms or $x^3$ terms. When teaching this concept, I often use the analogy of fruits. You can easily say you have 3 apples + 2 apples = 5 apples. But you can't directly add apples and bananas to get a single type of fruit. The same applies to algebraic terms. The ability to identify and combine like terms is crucial not only for simplifying expressions but also for solving equations and tackling more advanced mathematical problems later on. Remember, the goal is to make complex expressions as simple and manageable as possible, paving the way for easier calculations and a clearer understanding of the underlying mathematical relationships. Understanding the concept of like terms is a cornerstone in algebra. Once you grasp this idea, you'll find that many algebraic manipulations become much more straightforward. It's all about recognizing the common threads that tie terms together, allowing you to simplify and solve with greater ease and confidence.

Identifying Like Terms in the Expression

Okay, so let's look closely at our expression: $42 c^2+27 c^2-4 c^2-15 c^2-43 c^2$. What do you notice about each term? Well, each term contains $c^2$. That means every single term in this expression is a like term! This makes our job way easier. Since they all have the same variable part ($c^2$), we can combine all of them.

When identifying like terms, pay close attention to both the variable and its exponent. For instance, in the expression $5x^3 + 7x - 3x^3 + 2x^2 - 4x$, the like terms are $5x^3$ and $-3x^3$, as well as $7x$ and $-4x$. The term $2x^2$ stands alone because there are no other terms with $x$ raised to the power of 2. Spotting these similarities is key to simplifying the expression correctly. A common mistake is to combine terms that look similar but aren't exactly the same. For example, students might mistakenly try to combine $2x^2$ with $2x$, which is incorrect because the exponents are different. Always double-check that both the variable and the exponent match before combining terms. Another helpful tip is to rearrange the expression to group like terms together. This can make it visually easier to identify which terms can be combined. For example, you could rewrite the expression above as $5x^3 - 3x^3 + 2x^2 + 7x - 4x$. This rearrangement makes it clear which terms can be combined, reducing the likelihood of errors. Practice is essential for mastering the art of identifying like terms. The more you work with algebraic expressions, the quicker and more accurately you'll be able to spot like terms. Start with simple expressions and gradually work your way up to more complex ones. With consistent effort, you'll develop a keen eye for recognizing like terms, making algebraic simplification a breeze.

Combining the Like Terms

Now comes the fun part – actually combining the terms! Since all the terms are like terms (they all have $c^2$), we simply add or subtract their coefficients. Here's how it looks:

$42 c^2+27 c^2-4 c^2-15 c^2-43 c^2 = (42 + 27 - 4 - 15 - 43) c^2$

Now, let's do the arithmetic:

$42 + 27 = 69$ $69 - 4 = 65$ $65 - 15 = 50$ $50 - 43 = 7$

So, we have:

$(42 + 27 - 4 - 15 - 43) c^2 = 7 c^2$

That's it! The simplified expression is $7c^2$.

Combining like terms is like adding up all the items of the same type. Think of it as collecting all your marbles, then all your baseball cards, and so on. You're just grouping similar things together. When combining like terms, always pay attention to the signs (positive or negative) in front of each term. These signs will determine whether you add or subtract the coefficients. For example, in the expression $3x - 5x + 2x$, you would add $3x$ and $2x$ to get $5x$, then subtract $5x$ from $5x$ to get $0x$, which simplifies to 0. Another common mistake is to forget to carry the variable part along with the coefficient. Remember, you're not just adding or subtracting numbers; you're adding or subtracting terms. So, the variable part ($c^2$ in our case) must remain the same throughout the process. Double-checking your work is always a good idea, especially when dealing with multiple terms and signs. A simple way to check is to substitute a value for the variable in the original expression and in the simplified expression. If the results are the same, then you've likely combined the like terms correctly. With practice, combining like terms will become second nature. It's a fundamental skill that will serve you well in algebra and beyond. So keep practicing, and you'll become a master of algebraic simplification in no time!

Final Simplified Expression

So, after combining all the like terms in the original expression $42 c^2+27 c^2-4 c^2-15 c^2-43 c^2$, we arrive at the simplified expression:

$7c^2$

This is our final answer. We've taken a somewhat lengthy expression and condensed it down to its simplest form by combining the like terms. This not only makes the expression easier to read but also easier to work with in future calculations or algebraic manipulations.

Simplifying expressions is a crucial skill in mathematics, particularly in algebra. It allows us to reduce complex expressions into more manageable forms, making it easier to solve equations, analyze functions, and understand mathematical relationships. By combining like terms, we are essentially tidying up the expression, removing unnecessary clutter and revealing the underlying structure. This process not only simplifies calculations but also enhances our understanding of the expression's behavior. Moreover, simplifying expressions is not just a theoretical exercise; it has practical applications in various fields, including physics, engineering, economics, and computer science. In these fields, complex mathematical models are often used to describe real-world phenomena, and simplifying these models is essential for making predictions and drawing insights. For example, in physics, simplifying equations can help us understand the motion of objects or the behavior of electromagnetic fields. In engineering, it can help us design efficient structures or optimize the performance of electronic circuits. And in economics, it can help us analyze market trends or predict economic growth. Therefore, mastering the art of simplifying expressions is not just about manipulating symbols on paper; it's about developing a powerful tool that can be used to solve real-world problems and make meaningful contributions to various fields. So, keep practicing, keep exploring, and keep simplifying!

Conclusion

Alright, guys, we've successfully combined the like terms in the expression $42 c^2+27 c^2-4 c^2-15 c^2-43 c^2$ and simplified it to $7c^2$. Remember, the key is to identify those like terms and then either add or subtract their coefficients. Keep practicing, and you'll become a pro at simplifying algebraic expressions! This skill is super important for tackling more advanced math problems later on, so make sure you've got a solid grasp of it. You've got this!