Simplifying Algebraic Expressions A Step By Step Guide To (-20a²x² - 17ax) + (10a²x² - 15ax)
Introduction to Simplifying Algebraic Expressions
In mathematics, simplifying algebraic expressions is a fundamental skill. It involves reducing an expression to its simplest form without changing its value. This often means combining like terms, which are terms that have the same variables raised to the same powers. Mastering this skill is crucial for solving equations, understanding algebraic concepts, and tackling more complex mathematical problems. Simplifying expressions not only makes them easier to work with but also reveals their underlying structure and relationships. This article will provide a comprehensive guide on how to simplify the given expression, explaining each step in detail to ensure clarity and understanding.
The process of simplification typically involves several key steps. First, identify like terms within the expression. Like terms are terms that have the same variables raised to the same powers. For instance, in the expression -20a²x² - 17ax + 10a²x² - 15ax, the terms -20a²x² and 10a²x² are like terms because they both contain a²x². Similarly, -17ax and -15ax are like terms as they both contain ax. Once you've identified like terms, the next step is to combine them. This is done by adding or subtracting the coefficients of the like terms. The coefficient is the numerical part of a term; for example, in the term -20a²x², the coefficient is -20. When combining like terms, you add or subtract the coefficients while keeping the variable part the same. For example, to combine -20a²x² and 10a²x², you would add -20 and 10 to get -10, resulting in -10a²x². This process makes the expression more concise and easier to manage.
Understanding the properties of arithmetic is also essential for simplifying expressions. The associative property allows you to regroup terms without changing the value of the expression. For example, (a + b) + c is the same as a + (b + c). The commutative property allows you to change the order of terms without affecting the result; for instance, a + b is the same as b + a. The distributive property is crucial when dealing with expressions involving parentheses. It states that a(b + c) = ab + ac. By applying these properties, you can manipulate expressions to combine like terms effectively. Simplifying expressions is not just about getting to the final answer; it’s also about understanding the underlying mathematical principles and developing problem-solving skills. The ability to simplify expressions is a foundational skill that supports further study in algebra and beyond. By mastering these techniques, students can approach more complex mathematical problems with confidence and precision.
Step-by-Step Solution to Simplifying the Expression
To simplify the expression (-20a²x² - 17ax) + (10a²x² - 15ax), we will follow a step-by-step approach, ensuring each step is clear and easy to understand. This process involves identifying like terms and combining them to reduce the expression to its simplest form. By breaking down the solution into manageable steps, we can avoid errors and gain a deeper understanding of the simplification process.
Step 1: Remove Parentheses
The first step in simplifying the expression is to remove the parentheses. In this case, we have (-20a²x² - 17ax) + (10a²x² - 15ax). Since there is a plus sign between the two sets of parentheses, we can simply remove them without changing the signs of the terms inside. This is because adding a positive expression does not alter the signs of the terms within that expression. So, removing the parentheses gives us -20a²x² - 17ax + 10a²x² - 15ax. This step is crucial because it allows us to rearrange and combine like terms in the subsequent steps. By eliminating the parentheses, we make the expression more accessible for further simplification. Understanding this step is fundamental for simplifying more complex algebraic expressions, as it sets the stage for combining terms and reducing the expression to its simplest form.
Removing parentheses is a basic but essential technique in algebra. When there is a plus sign preceding a set of parentheses, the operation is straightforward: simply remove the parentheses. However, if there is a minus sign before the parentheses, it is crucial to distribute the negative sign to each term inside the parentheses. This means changing the sign of each term (positive to negative and negative to positive). For example, if we had -(a + b), it would become -a - b. In our case, because we have a plus sign, the removal is direct and doesn't require changing any signs. This simple step makes a significant difference in the clarity and manageability of the expression, paving the way for the next steps in the simplification process. Ensuring that this step is performed correctly is vital for achieving the correct final simplified expression.
Step 2: Identify Like Terms
The next crucial step in simplifying the expression is to identify like terms. Like terms are terms that have the same variables raised to the same powers. In our expression, -20a²x² - 17ax + 10a²x² - 15ax, we can identify two pairs of like terms. The first pair consists of -20a²x² and 10a²x². These terms are like terms because they both contain the variables a and x, each raised to the power of 2 for a and x in the first term, and the same for the second term. The second pair of like terms is -17ax and -15ax. These terms are like terms because they both contain the variables a and x, each raised to the power of 1 (which is implied when no exponent is written).
Identifying like terms is a fundamental skill in algebra, as it allows us to combine terms and simplify expressions. To effectively identify like terms, you need to carefully examine the variables and their exponents. Terms with different variables or the same variables raised to different powers are not like terms and cannot be combined directly. For example, a²x² and ax are not like terms because the exponents of a and x are different in each term. Recognizing like terms is essential for performing operations such as addition and subtraction correctly. This step sets the foundation for the next step, where we will combine these identified like terms to simplify the expression further. By accurately identifying like terms, we ensure that the subsequent steps lead to the correct simplified form of the expression. This skill is not only important for simplifying expressions but also for solving equations and other algebraic problems.
Step 3: Combine Like Terms
Now that we have identified the like terms in the expression -20a²x² - 17ax + 10a²x² - 15ax, the next step is to combine them. Combining like terms involves adding or subtracting the coefficients of the terms that have the same variables raised to the same powers. First, let’s combine the a²x² terms: -20a²x² + 10a²x². To do this, we add the coefficients -20 and 10, which gives us -10. Therefore, -20a²x² + 10a²x² = -10a²x². Next, we combine the ax terms: -17ax - 15ax. Here, we add the coefficients -17 and -15, which gives us -32. So, -17ax - 15ax = -32ax.
Combining like terms is a core concept in simplifying algebraic expressions. It allows us to reduce the complexity of an expression by consolidating terms that can be grouped together. The process involves adding or subtracting the numerical coefficients while keeping the variable part the same. For instance, when combining -20a²x² and 10a²x², we focus on the coefficients -20 and 10, performing the operation -20 + 10 = -10, and then retaining the a²x² part. Similarly, when combining -17ax and -15ax, we add the coefficients -17 and -15 to get -32, and keep the ax part. After combining all like terms, we have a simplified expression that is easier to understand and work with. This step is essential for solving equations, simplifying further expressions, and performing other algebraic manipulations. Accurate combination of like terms is crucial for achieving the correct final simplified form of the expression.
Step 4: Write the Simplified Expression
After combining the like terms, we can now write the simplified expression. We found that -20a²x² + 10a²x² simplifies to -10a²x², and -17ax - 15ax simplifies to -32ax. Therefore, the simplified expression is -10a²x² - 32ax. This expression is the most reduced form of the original expression, combining all like terms into single terms.
Writing the simplified expression is the final step in the simplification process. It represents the culmination of all the previous steps, where we identified like terms, combined their coefficients, and now present the result in its most concise form. The expression -10a²x² - 32ax is simpler than the original expression because it contains fewer terms and is easier to manage in further calculations or algebraic manipulations. This step demonstrates the power of simplification in making complex expressions more accessible and understandable. Ensuring that the final simplified expression is correctly written is crucial for the accuracy of any subsequent work that relies on this simplification. The ability to simplify expressions to their final form is a foundational skill in algebra and is essential for success in more advanced mathematical topics.
Final Simplified Expression
The final simplified expression for (-20a²x² - 17ax) + (10a²x² - 15ax) is -10a²x² - 32ax. This result is obtained by removing the parentheses, identifying like terms, and combining them. The steps involved in simplifying the expression are crucial for understanding and solving algebraic problems. This simplified form is not only more concise but also easier to work with in further calculations or algebraic manipulations. Understanding how to simplify expressions is a fundamental skill in algebra, and it provides a solid foundation for more advanced mathematical concepts.
The importance of achieving the final simplified expression cannot be overstated. It not only provides the most concise representation of the original expression but also makes it easier to identify key properties and relationships within the expression. The final simplified form -10a²x² - 32ax allows us to quickly see the terms and their coefficients, which can be invaluable for further analysis or problem-solving. This skill is essential in various fields, including engineering, physics, and computer science, where complex expressions often need to be simplified for practical applications. Mastering the process of simplification equips students and professionals with a powerful tool for tackling mathematical challenges with confidence and precision.
Conclusion
In conclusion, simplifying the expression (-20a²x² - 17ax) + (10a²x² - 15ax) involves a series of fundamental steps, ultimately leading to the simplified form -10a²x² - 32ax. This process highlights the importance of identifying and combining like terms, a crucial skill in algebra. By following these steps, we can transform complex expressions into simpler, more manageable forms. The ability to simplify expressions is not only essential for solving equations but also for understanding and manipulating mathematical concepts in various fields.
Understanding the process of simplifying algebraic expressions is a cornerstone of mathematical proficiency. It allows for a clearer understanding of mathematical relationships and makes it easier to work with complex problems. The steps outlined in this article—removing parentheses, identifying like terms, combining them, and writing the simplified expression—provide a systematic approach to simplification. This skill is not just for academic purposes; it has practical applications in various real-world scenarios, from engineering and finance to computer programming and data analysis. By mastering these techniques, individuals can enhance their problem-solving abilities and approach mathematical challenges with greater confidence and efficiency. The simplified form -10a²x² - 32ax is a testament to the power of these algebraic manipulations, showcasing how a seemingly complex expression can be reduced to its most essential components.