Step By Step Guide To Simplify $6q^2 - 7q^2 + 9q^2 + 5q^2 - 2q^2 - 2q^2$

Introduction: Mastering Algebraic Simplification

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex equations and reduce them to their most basic form, making them easier to understand and manipulate. This article delves into the process of simplifying the algebraic expression $6q^2 - 7q^2 + 9q^2 + 5q^2 - 2q^2 - 2q^2$. We'll break down the steps involved, providing a clear and concise explanation that will benefit students, educators, and anyone looking to brush up on their algebra skills. Understanding how to simplify algebraic expressions is crucial not only for academic success but also for various real-world applications, such as problem-solving in physics, engineering, and computer science. Before we dive into the specifics of this expression, let's briefly review the basic principles of algebraic simplification. The key concept here is combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, $3x^2$ and $5x^2$ are like terms, while $3x^2$ and $5x$ are not. When simplifying, we can only add or subtract like terms. This is because we are essentially combining quantities of the same item. Think of it like adding apples and apples versus adding apples and oranges; you can add apples together to get a total number of apples, but you cannot directly add apples and oranges. Similarly, we can combine terms with $q^2$ but not terms with different powers of $q$. With this foundation in place, let's embark on our journey to simplify the given expression. We'll take a methodical approach, ensuring each step is clearly explained. Our goal is not just to arrive at the correct answer but to understand the underlying principles that make algebraic simplification possible. This understanding will empower you to tackle a wide range of algebraic problems with confidence and precision. Remember, practice is key in mathematics. The more you practice simplifying expressions, the more natural and intuitive the process will become. So, grab a pen and paper, and let's get started on simplifying this expression together. By the end of this article, you'll have a solid grasp of how to simplify algebraic expressions, and you'll be well-equipped to tackle more complex problems in the future.

Step 1: Identifying Like Terms

The first crucial step in simplifying the algebraic expression $6q^2 - 7q^2 + 9q^2 + 5q^2 - 2q^2 - 2q^2$ is to identify the like terms. In this case, all the terms are like terms because each term contains the variable $q$ raised to the power of 2 (i.e., $q^2$). This makes our task significantly easier, as we can directly combine all the terms. If the expression had contained terms with different powers of $q$, such as $q$ or $q^3$, we would only combine the terms with the same power. For example, if we had the expression $6q^2 - 7q + 9q^2 + 5q - 2q^2 - 2q$, we would group the $q^2$ terms together ($6q^2$, $9q^2$, and $-2q^2$) and the $q$ terms together ($-7q$, $5q$, and $-2q$). However, in our current expression, we are fortunate to have all like terms, which simplifies the process. Recognizing like terms is a fundamental skill in algebra. It's the foundation upon which we build more complex simplification techniques. Think of it like sorting objects into categories; you can only combine objects that belong to the same category. In algebra, the categories are defined by the variables and their exponents. Once you've mastered the art of identifying like terms, the rest of the simplification process becomes much smoother. This step also helps in preventing common errors, such as incorrectly combining unlike terms. For instance, a common mistake is to try to add $q^2$ and $q$ terms directly, which is mathematically incorrect. By focusing on identifying like terms first, we set ourselves up for success in the subsequent steps. Now that we've established that all terms in our expression are like terms, we can proceed to the next step, which involves combining these terms through addition and subtraction. This is where the arithmetic comes into play, and we'll carefully work through the operations to arrive at the simplified form of the expression.

Step 2: Combining Like Terms

Now that we have identified that all terms in the expression $6q^2 - 7q^2 + 9q^2 + 5q^2 - 2q^2 - 2q^2$ are like terms, the next step is to combine them. This involves performing the arithmetic operations (addition and subtraction) on the coefficients of the like terms. The coefficient is the numerical factor in a term. For instance, in the term $6q^2$, the coefficient is 6. To combine like terms, we simply add or subtract the coefficients while keeping the variable part ($q^2$ in this case) the same. Let's start by combining the first two terms: $6q^2 - 7q^2$. This is equivalent to $(6 - 7)q^2$, which simplifies to $-1q^2$ or simply $-q^2$. Next, we add the third term: $-q^2 + 9q^2$. This is equivalent to $(-1 + 9)q^2$, which simplifies to $8q^2$. We continue this process, adding the fourth term: $8q^2 + 5q^2$. This is equivalent to $(8 + 5)q^2$, which simplifies to $13q^2$. Now, we subtract the fifth term: $13q^2 - 2q^2$. This is equivalent to $(13 - 2)q^2$, which simplifies to $11q^2$. Finally, we subtract the last term: $11q^2 - 2q^2$. This is equivalent to $(11 - 2)q^2$, which simplifies to $9q^2$. Therefore, after combining all the like terms, the simplified expression is $9q^2$. This step-by-step approach ensures accuracy and helps in understanding the process. It's important to pay close attention to the signs (positive or negative) of the coefficients, as an error in sign can lead to an incorrect result. Combining like terms is a fundamental operation in algebra and is used extensively in solving equations, simplifying expressions, and working with polynomials. Mastering this skill is crucial for success in higher-level mathematics. By carefully performing the arithmetic operations on the coefficients, we can reduce complex expressions to their simplest form, making them easier to work with and understand.

Step 3: The Simplified Expression

After carefully combining all the like terms in the expression $6q^2 - 7q^2 + 9q^2 + 5q^2 - 2q^2 - 2q^2$, we have arrived at the simplified expression, which is $9q^2$. This is the most concise and basic form of the original expression. The process of simplification has allowed us to reduce six terms into a single term, making the expression easier to interpret and use in further calculations. The simplified form $9q^2$ represents the same value as the original expression for any given value of $q$. This is a crucial concept in algebra: simplification does not change the value of the expression; it merely represents it in a more manageable form. To illustrate this, let's consider a few examples. If we substitute $q = 2$ into the original expression, we get: $6(2)^2 - 7(2)^2 + 9(2)^2 + 5(2)^2 - 2(2)^2 - 2(2)^2 = 6(4) - 7(4) + 9(4) + 5(4) - 2(4) - 2(4) = 24 - 28 + 36 + 20 - 8 - 8 = 36$. If we substitute $q = 2$ into the simplified expression, we get: $9(2)^2 = 9(4) = 36$. As we can see, both expressions yield the same result, demonstrating that simplification preserves the value of the expression. Similarly, if we substitute $q = -1$ into the original expression, we get: $6(-1)^2 - 7(-1)^2 + 9(-1)^2 + 5(-1)^2 - 2(-1)^2 - 2(-1)^2 = 6(1) - 7(1) + 9(1) + 5(1) - 2(1) - 2(1) = 6 - 7 + 9 + 5 - 2 - 2 = 9$. And substituting $q = -1$ into the simplified expression, we get: $9(-1)^2 = 9(1) = 9$. Again, both expressions give the same result. The simplified expression $9q^2$ is not only more compact but also makes it easier to perform further operations, such as evaluating the expression for different values of $q$, graphing the expression, or using it in more complex equations.

Conclusion: The Power of Simplification

In conclusion, we have successfully simplified the algebraic expression $6q^2 - 7q^2 + 9q^2 + 5q^2 - 2q^2 - 2q^2$ to its most basic form: $9q^2$. This process highlights the power of simplification in mathematics, allowing us to transform complex expressions into more manageable forms. The key steps involved were identifying like terms and then combining them through addition and subtraction. These are fundamental skills in algebra that are essential for solving equations, working with polynomials, and tackling a wide range of mathematical problems. Understanding how to simplify expressions is not just an academic exercise; it's a crucial tool for problem-solving in various fields, including engineering, physics, computer science, and economics. Simplified expressions are easier to understand, easier to evaluate, and easier to use in further calculations. They provide a clear and concise representation of the underlying mathematical relationship. Throughout this article, we emphasized a step-by-step approach, breaking down the simplification process into manageable steps. This methodical approach helps prevent errors and promotes a deeper understanding of the underlying concepts. Remember, mathematics is not just about finding the right answer; it's about understanding the process and reasoning behind the solution. By mastering the art of simplification, you'll be well-equipped to tackle more complex mathematical challenges. The ability to simplify expressions efficiently and accurately is a valuable asset in any mathematical endeavor. It allows you to focus on the core concepts of the problem rather than getting bogged down in complex calculations. So, continue to practice simplifying expressions, and you'll find that your algebraic skills will grow stronger with each problem you solve. The journey through this example illustrates the elegance and efficiency of mathematical simplification, a skill that will serve you well in your future mathematical pursuits. The simplified expression $9q^2$ is not just an answer; it's a testament to the power of algebraic manipulation and a stepping stone to more advanced mathematical concepts.