Simplifying Algebraic Expressions A Step-by-Step Guide To 5x + 4 - 9y + 3x + 2y - 7

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Understanding Algebraic Expressions

In mathematics, simplifying algebraic expressions is a fundamental skill. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. To simplify an algebraic expression means to rewrite it in its most compact form, while retaining its original value. This typically involves combining like terms, which are terms that have the same variable raised to the same power. Mastering the simplification of algebraic expressions is crucial as it lays the groundwork for solving more complex equations and problems in algebra and other branches of mathematics. The ability to efficiently and accurately simplify expressions is not only essential for academic success but also for practical applications in various fields, including engineering, physics, and computer science. By understanding the underlying principles and techniques, one can confidently tackle a wide range of algebraic challenges. Simplifying algebraic expressions helps to make mathematical problems more manageable and easier to solve, allowing for a clearer understanding of the relationships between variables and constants. It is a cornerstone of algebraic manipulation, enabling us to work with mathematical models and formulas effectively. Therefore, a solid grasp of this concept is invaluable for anyone pursuing studies or careers that involve mathematical reasoning and problem-solving.

Problem Statement

Our primary task is to simplify the given algebraic expression: 5x + 4 - 9y + 3x + 2y - 7. This expression contains terms with variables x and y, as well as constant terms. To simplify it, we need to identify and combine like terms. This involves grouping terms with the same variable and combining their coefficients, as well as combining the constant terms. The process of simplification is not just about reducing the number of terms; it's about making the expression easier to understand and work with. By simplifying, we can reveal the underlying structure of the expression and make it more amenable to further mathematical operations. This skill is particularly important in various areas of mathematics, such as solving equations, graphing functions, and calculus. The ability to simplify expressions accurately and efficiently is a hallmark of mathematical proficiency. In this specific problem, we will apply the principles of combining like terms to reduce the expression to its simplest form, demonstrating a key concept in algebra. The simplified expression will be mathematically equivalent to the original but will be more concise and easier to interpret. This exercise underscores the importance of simplification in algebraic manipulation and its role in enhancing mathematical problem-solving skills.

Step-by-Step Solution

1. Identify Like Terms

The first step in simplifying the algebraic expression 5x + 4 - 9y + 3x + 2y - 7 is to identify like terms. Like terms are those that contain the same variable raised to the same power. In this expression, we have two terms with the variable x (5x and 3x), two terms with the variable y (-9y and 2y), and two constant terms (4 and -7). Grouping these like terms together is crucial for the next step. This process of identifying and grouping is fundamental to simplifying any algebraic expression. Without correctly identifying like terms, the subsequent steps will lead to an incorrect simplification. The ability to recognize like terms quickly and accurately is a key skill in algebra. This step lays the foundation for combining the terms and reducing the expression to its simplest form. By systematically identifying and grouping like terms, we ensure that we are only combining terms that can be mathematically combined, thus maintaining the integrity of the expression. This careful approach is essential for achieving a correct and simplified result. Recognizing like terms is not just about spotting the same variable; it's about understanding the structure of the algebraic expression and the relationships between its components.

2. Group Like Terms

Once we've identified the like terms, the next step is to group them together. This involves rearranging the expression to bring the like terms into proximity. For the expression 5x + 4 - 9y + 3x + 2y - 7, we can group the terms as follows: (5x + 3x) + (-9y + 2y) + (4 - 7). Grouping like terms makes it visually clearer which terms can be combined. This step is crucial for avoiding errors in the simplification process. By physically grouping the terms, we create a structured framework for the next operation, which is combining the coefficients. This systematic approach ensures that each term is accounted for and that the simplification process is carried out in an organized manner. Grouping is not just about visual clarity; it's about creating a mathematical structure that facilitates accurate calculation. This step helps to break down the complex expression into smaller, more manageable parts, making it easier to simplify. The grouped terms act as individual units that can be processed separately and then combined to form the simplified expression. This methodical approach is essential for handling more complex algebraic expressions with multiple variables and terms.

3. Combine Like Terms

Now that we have grouped the like terms, we can proceed to combine them. This involves adding or subtracting the coefficients of the like terms. In our grouped expression (5x + 3x) + (-9y + 2y) + (4 - 7), we combine the coefficients as follows:

  • 5x + 3x = 8x
  • -9y + 2y = -7y
  • 4 - 7 = -3

Combining like terms is a fundamental operation in simplifying algebraic expressions. It involves understanding the arithmetic operations and applying them correctly to the coefficients of the variables and the constant terms. This step is crucial for reducing the expression to its simplest form. The process of combining like terms not only reduces the number of terms in the expression but also makes the expression easier to understand and work with. Accuracy in this step is paramount, as any error in combining the coefficients will lead to an incorrect simplification. This step demonstrates the power of algebraic manipulation in transforming a complex expression into a more manageable one. By combining like terms, we reveal the underlying structure of the expression and make it more amenable to further mathematical operations. This skill is essential for solving equations, graphing functions, and other advanced mathematical concepts.

4. Write the Simplified Expression

After combining the like terms, we write the simplified expression by joining the results together. From the previous step, we found that:

  • 5x + 3x = 8x
  • -9y + 2y = -7y
  • 4 - 7 = -3

Therefore, the simplified expression is 8x - 7y - 3. This is the most compact form of the original expression, where all like terms have been combined. Writing the simplified expression is the final step in the simplification process. It represents the culmination of all the previous steps, where we identified, grouped, and combined like terms. The simplified expression is mathematically equivalent to the original but is easier to understand and work with. This form is crucial for solving equations, graphing functions, and performing other algebraic manipulations. The ability to simplify expressions and present them in their simplest form is a key skill in mathematics. It demonstrates a clear understanding of algebraic principles and the ability to apply them effectively. The simplified expression provides a clear and concise representation of the original expression, making it easier to analyze and interpret. This final step underscores the importance of simplification in making mathematical problems more manageable and accessible.

Final Answer

The simplified form of the algebraic expression 5x + 4 - 9y + 3x + 2y - 7 is 8x - 7y - 3. This result is achieved by identifying and combining like terms, a fundamental technique in algebraic simplification. The ability to simplify expressions is crucial for solving more complex mathematical problems and is a cornerstone of algebraic manipulation. By reducing the expression to its simplest form, we make it easier to understand and work with. This final answer represents the culmination of the step-by-step solution, where we methodically grouped and combined like terms to arrive at the most concise representation of the original expression. The simplified form not only reduces the number of terms but also reveals the underlying structure of the expression, making it more amenable to further mathematical operations. This exercise highlights the importance of simplification in mathematics and its role in enhancing problem-solving skills. The final answer, 8x - 7y - 3, is a clear and concise representation of the original expression, demonstrating the power of algebraic simplification.