Distributive Property And Combining Like Terms A Step-by-Step Guide

Simplify the expression x+3y-y+3x+2(2+4+y) using the Distributive Property and combining like terms. Which of the following is the correct result A 5x+8y+15 B 4x+4y+12 C 4x+3y+12

In mathematics, the distributive property is a fundamental concept that allows us to simplify expressions involving parentheses. It essentially states that multiplying a number by a sum or difference is the same as multiplying the number by each term inside the parentheses individually and then adding or subtracting the results. This property is crucial for manipulating algebraic expressions and solving equations. In this comprehensive guide, we will delve into the intricacies of the distributive property and explore how it can be applied to simplify complex expressions, combine like terms, and ultimately enhance your mathematical problem-solving skills. Mastering this property is essential for success in algebra and beyond, as it lays the groundwork for more advanced mathematical concepts.

To effectively grasp the distributive property, it's important to understand its core principle. When we encounter an expression in the form of a(b + c), the distributive property allows us to rewrite it as ab + ac. This means we multiply the term outside the parentheses (a) by each term inside the parentheses (b and c) and then add the products together. Similarly, for an expression like a(b - c), the distributive property transforms it into ab - ac. The key is to distribute the term outside the parentheses to every term within, paying close attention to the signs. This seemingly simple concept is the foundation for simplifying more intricate expressions and solving equations. By understanding and applying the distributive property, you can break down complex problems into manageable steps, making the process of algebraic manipulation much more efficient and accurate.

Let's consider a practical example to illustrate the distributive property in action. Imagine we have the expression 3(x + 2). To apply the distributive property, we multiply 3 by both x and 2. This gives us 3 * x + 3 * 2, which simplifies to 3x + 6. This transformation demonstrates how the distributive property allows us to eliminate parentheses and express the original expression in a more simplified form. This is particularly useful when dealing with expressions that contain multiple terms within parentheses or when solving equations where parentheses hinder direct calculations. By mastering this technique, you gain a powerful tool for manipulating algebraic expressions and solving a wide range of mathematical problems. The distributive property not only simplifies calculations but also provides a clear and logical approach to algebraic manipulations, making it an indispensable skill for any aspiring mathematician or problem-solver.

After applying the distributive property, the next crucial step in simplifying expressions is combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, 3x and 5x are like terms because they both have the variable 'x' raised to the power of 1. Similarly, 2y² and -7y² are like terms because they both have the variable 'y' raised to the power of 2. Constant terms, such as 4 and -9, are also considered like terms. Combining like terms involves adding or subtracting their coefficients (the numbers in front of the variables) while keeping the variable and its exponent the same. This process streamlines expressions, making them easier to understand and work with. In essence, combining like terms is about grouping similar elements together to simplify the overall expression.

To effectively combine like terms, it's essential to first identify them within an expression. This often involves carefully examining each term and comparing its variable and exponent to those of other terms. Once like terms are identified, you can proceed to add or subtract their coefficients. For example, in the expression 4x + 2y - x + 5y, the like terms are 4x and -x, as well as 2y and 5y. Combining 4x and -x gives us 3x, while combining 2y and 5y gives us 7y. Therefore, the simplified expression becomes 3x + 7y. This process not only reduces the number of terms in the expression but also makes it easier to interpret and manipulate. By mastering the technique of combining like terms, you can significantly simplify algebraic expressions and lay the groundwork for solving more complex equations and problems. This skill is fundamental to algebraic proficiency and is widely used in various mathematical contexts.

Consider the expression 7a² - 3a + 2a² + 5a - 1. To combine like terms, we first identify the terms with the same variable and exponent. The like terms are 7a² and 2a², as well as -3a and 5a. Combining 7a² and 2a² gives us 9a², while combining -3a and 5a gives us 2a. The constant term -1 remains unchanged as there are no other constant terms to combine with. Therefore, the simplified expression is 9a² + 2a - 1. This example highlights the importance of carefully identifying and grouping like terms before performing any arithmetic operations. By systematically combining like terms, we can transform complex expressions into simpler, more manageable forms, making them easier to analyze and use in further calculations. This skill is a cornerstone of algebraic manipulation and is essential for success in higher-level mathematics.

Now, let's tackle the original problem step-by-step, applying the distributive property and combining like terms to arrive at the correct answer. The expression we need to simplify is: $x+3y-y+3x+2(2+4+y)$. This expression presents a combination of terms and a set of parentheses that require our attention. By systematically applying the distributive property and combining like terms, we can transform this complex expression into a simplified form. This process not only demonstrates the practical application of these mathematical concepts but also reinforces the importance of a step-by-step approach to problem-solving. By carefully breaking down the problem into manageable steps, we can avoid errors and arrive at the correct solution with confidence.

Step 1: Apply the Distributive Property. The first step is to distribute the 2 in the term 2(2 + 4 + y). This means we multiply 2 by each term inside the parentheses: 2 * 2 = 4, 2 * 4 = 8, and 2 * y = 2y. So, the expression becomes: $x + 3y - y + 3x + 4 + 8 + 2y$. This step is crucial as it eliminates the parentheses and allows us to combine like terms in the subsequent steps. The distributive property is a powerful tool for simplifying expressions, and its application here sets the stage for further simplification. By understanding and applying this property, we can effectively handle expressions that contain parentheses and pave the way for solving more complex algebraic problems.

Step 2: Combine Like Terms. Next, we identify and combine the like terms. We have the following like terms: x and 3x, 3y, -y, and 2y, and the constants 4 and 8. Combining x and 3x gives us 4x. Combining 3y, -y, and 2y gives us 4y. Combining the constants 4 and 8 gives us 12. Therefore, the expression simplifies to: $4x + 4y + 12$. This step demonstrates the power of combining like terms to streamline expressions. By grouping similar terms together, we reduce the complexity of the expression and make it easier to understand and work with. This skill is fundamental to algebraic manipulation and is essential for solving equations and simplifying more complex mathematical problems.

Therefore, the correct answer is B. 4x + 4y + 12. By carefully applying the distributive property and combining like terms, we have successfully simplified the given expression and arrived at the correct solution. This process highlights the importance of a systematic approach to problem-solving in mathematics. By breaking down complex problems into smaller, manageable steps, we can avoid errors and arrive at the correct answer with confidence. This example serves as a valuable illustration of how the distributive property and combining like terms can be used to simplify algebraic expressions and solve mathematical problems effectively.

In conclusion, mastering the distributive property and the ability to combine like terms are essential skills in algebra. These techniques allow us to simplify complex expressions and solve equations effectively. By understanding and applying these concepts, you can enhance your mathematical problem-solving abilities and achieve greater success in your mathematical journey. The distributive property serves as a fundamental tool for manipulating expressions containing parentheses, while combining like terms streamlines expressions by grouping similar elements together. These skills are not only crucial for algebraic manipulations but also lay the groundwork for more advanced mathematical concepts. By consistently practicing and applying these techniques, you can develop a strong foundation in algebra and confidently tackle a wide range of mathematical problems.