Simplifying Algebraic Expressions -9x² - 2x + 7 + 6x² - 8x + 1: A Step-by-Step Guide

Introduction to Algebraic Expressions

Algebraic expressions form the bedrock of algebra, serving as a gateway to more advanced mathematical concepts. Mastering the simplification of these expressions is crucial for success in algebra and beyond. In essence, an algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables, often represented by letters like x, y, or z, stand for unknown values, while constants are fixed numerical values. Simplifying an algebraic expression involves rearranging and combining like terms to present the expression in its most concise and manageable form. This process not only makes the expression easier to understand but also facilitates further calculations and problem-solving.

At its core, simplification hinges on the concept of like terms. Like terms are those that have the same variable raised to the same power. For example, 3x² and -5x² are like terms because they both contain the variable x raised to the power of 2. Similarly, 7x and 2x are like terms. Constants, such as 4 and -9, are also considered like terms. The golden rule of simplification is that only like terms can be combined through addition or subtraction. This principle ensures that we are not mixing apples and oranges, so to speak, and that the resulting expression remains mathematically equivalent to the original.

Why is simplifying algebraic expressions so important? The answer lies in its practical applications. Simplified expressions are easier to evaluate, meaning that when we substitute specific values for the variables, the calculations become less cumbersome. This is particularly important in real-world scenarios where algebraic expressions are used to model various phenomena. Moreover, simplified expressions are easier to manipulate, which is essential when solving equations or inequalities. A simplified expression allows us to isolate the variable of interest more efficiently, leading to a solution more quickly and accurately. In essence, simplification is a tool that enhances our ability to work with algebraic expressions, making them more accessible and useful.

Understanding the Given Expression: -9x² - 2x + 7 + 6x² - 8x + 1

To effectively simplify an algebraic expression, the first crucial step is to thoroughly understand the expression at hand. In this particular case, we are presented with the expression -9x² - 2x + 7 + 6x² - 8x + 1. At first glance, it may appear complex, but breaking it down into its constituent parts reveals its underlying structure. This expression comprises several terms, each separated by addition or subtraction signs. These terms can be broadly classified into three categories: quadratic terms (terms involving x²), linear terms (terms involving x), and constant terms (numerical values without any variables).

Let's delve deeper into each category. The quadratic terms in this expression are -9x² and 6x². These terms are characterized by the variable x being raised to the power of 2. The coefficients, which are the numerical factors multiplying the variable part, are -9 and 6, respectively. The linear terms are -2x and -8x. These terms involve the variable x raised to the power of 1 (although the exponent is not explicitly written). The coefficients of these linear terms are -2 and -8. Finally, the constant terms are 7 and 1. These terms are simply numerical values without any variable component.

Identifying these components is essential for the next step in the simplification process: grouping like terms. By recognizing the quadratic, linear, and constant terms, we can strategically rearrange the expression to bring like terms together. This rearrangement is permissible due to the commutative property of addition, which states that the order in which terms are added does not affect the sum. Grouping like terms is a crucial step because it sets the stage for combining them, which is the core of the simplification process. Without a clear understanding of the expression's components and the ability to identify like terms, the simplification process can become confusing and prone to errors. Therefore, taking the time to dissect the expression and recognize its individual parts is a worthwhile investment in ensuring accurate and efficient simplification.

Step-by-Step Simplification Process

The core of simplifying algebraic expressions lies in the methodical process of identifying and combining like terms. This step-by-step approach ensures accuracy and clarity, transforming a potentially complex expression into a more manageable form. In this section, we will break down the simplification of the given expression, -9x² - 2x + 7 + 6x² - 8x + 1, into easily digestible steps.

1. Grouping Like Terms

The first critical step is to group like terms together. This involves rearranging the expression so that terms with the same variable and exponent are placed next to each other. As discussed earlier, like terms are those that have the same variable raised to the same power. In our expression, we have quadratic terms (-9x² and 6x²), linear terms (-2x and -8x), and constant terms (7 and 1). Using the commutative property of addition, we can rearrange the expression as follows:

-9x² + 6x² - 2x - 8x + 7 + 1

By grouping like terms in this manner, we create a visual organization that makes it easier to identify and combine terms in the subsequent steps. This rearrangement is not mathematically altering the expression; it is merely presenting it in a way that facilitates the simplification process.

2. Combining Like Terms

The next step is to combine the like terms that we have grouped together. This involves performing the indicated operations (addition or subtraction) on the coefficients of the like terms. Remember, only like terms can be combined. Let's start with the quadratic terms: -9x² + 6x². To combine these terms, we add their coefficients: -9 + 6 = -3. Therefore, -9x² + 6x² simplifies to -3x².

Next, we move on to the linear terms: -2x - 8x. Again, we combine the coefficients: -2 - 8 = -10. Thus, -2x - 8x simplifies to -10x.

Finally, we combine the constant terms: 7 + 1. This is a simple addition: 7 + 1 = 8.

3. Writing the Simplified Expression

After combining all the like terms, the final step is to write the simplified expression. This involves arranging the simplified terms in a standard form, typically in descending order of the exponent of the variable. In our case, we have the simplified terms -3x², -10x, and 8. Arranging them in descending order of the exponent of x gives us:

-3x² - 10x + 8

This is the fully simplified form of the original expression. It is more concise and easier to work with compared to the original expression. By following these steps meticulously, we can confidently simplify any algebraic expression, regardless of its complexity.

Final Simplified Expression: -3x² - 10x + 8

In the previous section, we meticulously walked through the step-by-step process of simplifying the algebraic expression -9x² - 2x + 7 + 6x² - 8x + 1. We began by grouping like terms, carefully rearranging the expression to bring together terms with the same variable and exponent. This strategic rearrangement, guided by the commutative property of addition, laid the groundwork for the next crucial step: combining like terms. We then systematically combined the coefficients of the quadratic terms, linear terms, and constant terms, adhering to the fundamental principle that only like terms can be added or subtracted. This process yielded the simplified terms -3x², -10x, and 8.

Finally, we assembled these simplified terms into a cohesive expression, arranging them in the standard form of descending order of the exponent of the variable. This convention ensures that the expression is presented in a clear and easily recognizable format. The culmination of this process is the final simplified expression: -3x² - 10x + 8. This expression is mathematically equivalent to the original expression, but it is significantly more concise and easier to interpret.

The importance of this simplified form cannot be overstated. It serves as a cornerstone for further algebraic manipulations, such as solving equations, graphing functions, and performing more complex calculations. Imagine trying to solve an equation containing the original expression -9x² - 2x + 7 + 6x² - 8x + 1. The complexity of the expression would likely introduce challenges and increase the potential for errors. However, by working with the simplified expression -3x² - 10x + 8, the process becomes significantly more streamlined and manageable. The simplified form provides a clear and direct representation of the relationship between the variables and constants, making it easier to apply algebraic techniques and arrive at accurate solutions.

Furthermore, the simplified expression offers a more intuitive understanding of the underlying mathematical relationship. The coefficients and terms in the simplified expression directly reveal the key characteristics of the expression, such as the degree of the polynomial (in this case, 2, indicating a quadratic expression) and the constant term (which represents the y-intercept when graphed). This clarity is invaluable for both conceptual understanding and practical application. In essence, the final simplified expression -3x² - 10x + 8 is not merely a cosmetic change; it is a powerful tool that enhances our ability to work with and understand algebraic relationships.

Conclusion

In conclusion, the process of simplifying algebraic expressions is a fundamental skill in algebra, with far-reaching implications for mathematical problem-solving and understanding. In this article, we have explored the step-by-step approach to simplifying the expression -9x² - 2x + 7 + 6x² - 8x + 1, demonstrating the core principles and techniques involved. We began by emphasizing the importance of understanding algebraic expressions and identifying like terms, which are the building blocks of simplification. We then meticulously walked through the process of grouping like terms, combining them through addition and subtraction, and writing the final simplified expression in a standard form.

The resulting expression, -3x² - 10x + 8, showcases the power of simplification. It is mathematically equivalent to the original expression, yet it is more concise, manageable, and easier to interpret. This simplification not only reduces the complexity of the expression but also facilitates further algebraic manipulations, such as solving equations and graphing functions. A simplified expression provides a clearer picture of the underlying mathematical relationship, making it easier to apply appropriate techniques and arrive at accurate solutions.

The ability to simplify algebraic expressions is not merely an academic exercise; it is a valuable skill that extends to various real-world applications. From modeling physical phenomena to optimizing business processes, algebraic expressions are used to represent and analyze a wide range of situations. By mastering the art of simplification, we empower ourselves to work with these expressions more effectively, unlocking insights and solutions that would otherwise remain hidden in the complexity of the original form.

The key takeaways from this discussion are the importance of understanding the structure of algebraic expressions, the necessity of identifying and grouping like terms, and the methodical process of combining these terms to arrive at a simplified form. By embracing these principles and practicing the techniques outlined in this article, anyone can develop the confidence and competence to simplify algebraic expressions with ease and accuracy. The journey through algebra is paved with simplification, and mastering this skill is a crucial step towards mathematical fluency and problem-solving prowess.