Simplifying Expressions Remove Parentheses And Collect Like Terms

by Admin 66 views

In mathematics, simplifying expressions is a fundamental skill that lays the groundwork for more advanced concepts. Often, expressions contain parentheses and like terms, which can make them appear complex. However, by mastering the techniques of removing parentheses and combining like terms, we can significantly simplify these expressions, making them easier to understand and manipulate. This article delves into the process of simplifying the expression 35p - (12p + 6), providing a step-by-step guide and explaining the underlying mathematical principles.

Understanding the order of operations is very important. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. When simplifying expressions, we must adhere to this order to arrive at the correct result. In the expression 35p - (12p + 6), we first address the parentheses. The presence of parentheses indicates that the operations within them should be performed before any operations outside them. This might involve simplifying the terms inside the parentheses or, as in this case, distributing a factor across the terms.

Removing parentheses is a crucial step in simplifying expressions. Parentheses often group terms together, indicating that they should be treated as a single unit. However, to further simplify the expression, we need to remove these parentheses. The method for removing parentheses depends on what precedes them. If a plus sign (+) precedes the parentheses, we can simply remove them without changing the signs of the terms inside. For example, a + (b + c) simplifies to a + b + c. However, if a minus sign (-) precedes the parentheses, we must distribute the negative sign to each term inside, effectively changing their signs. This is because subtracting a group of terms is the same as adding the negative of each term. In our expression, 35p - (12p + 6), we have a minus sign before the parentheses, so we need to distribute it.

Let's break down the simplification of the expression 35p - (12p + 6) into manageable steps:

  1. Distribute the negative sign: The first step is to distribute the negative sign in front of the parentheses to each term inside. This means we multiply each term inside the parentheses by -1. So, -(12p + 6) becomes -12p - 6. Remember that when we multiply a positive term by a negative number, the result is negative, and when we multiply a positive term by a negative number, the result is negative. This is a fundamental rule of arithmetic that is essential for correctly distributing the negative sign.

  2. Rewrite the expression: After distributing the negative sign, we rewrite the expression as 35p - 12p - 6. Notice that the parentheses are now gone, and the expression is a series of terms that are either added or subtracted. This is a crucial step because it allows us to combine the like terms in the next step. The expression is now in a form that is much easier to work with, and we can proceed with the simplification process.

  3. Identify like terms: Like terms are terms that have the same variable raised to the same power. In our expression, 35p and -12p are like terms because they both have the variable p raised to the power of 1. The term -6 is a constant term and does not have any variable, so it is not a like term with 35p and -12p. Identifying like terms is a critical step in simplifying expressions because only like terms can be combined. Combining like terms allows us to reduce the number of terms in the expression and make it more concise.

  4. Combine like terms: To combine like terms, we add or subtract their coefficients (the numbers in front of the variables). In this case, we have 35p - 12p. To combine these terms, we subtract the coefficients: 35 - 12 = 23. So, 35p - 12p simplifies to 23p. This step is the heart of simplifying expressions because it reduces the complexity of the expression by combining similar terms. The result, 23p, is a single term that represents the combined value of the original two terms.

  5. Write the simplified expression: After combining the like terms, we write the simplified expression. In this case, we have 23p - 6. This is the simplest form of the original expression, as there are no more like terms to combine and no parentheses to remove. The simplified expression is much easier to work with and understand than the original expression, which is the ultimate goal of simplification.

Like terms are the cornerstone of simplifying algebraic expressions. They are terms that share the same variable raised to the same power. For example, 3x and 5x are like terms because they both have the variable x raised to the power of 1. Similarly, 2y^2 and -7y^2 are like terms because they both have the variable y raised to the power of 2. However, 4x and 4x^2 are not like terms because they have the same variable (x) but raised to different powers (1 and 2, respectively). Similarly, 3x and 3y are not like terms because they have different variables (x and y, respectively).

Combining like terms involves adding or subtracting their coefficients while keeping the variable and its exponent the same. The coefficient is the numerical part of the term. For instance, in the term 5x, the coefficient is 5. To combine like terms, we simply add or subtract their coefficients. For example, to combine 3x + 5x, we add the coefficients 3 and 5, resulting in 8x. Similarly, to combine 7y^2 - 2y^2, we subtract the coefficients 2 from 7, resulting in 5y^2. The variable and its exponent remain unchanged during this process. It's crucial to remember that only like terms can be combined; we cannot combine terms with different variables or different exponents.

The ability to identify and combine like terms is essential for simplifying expressions and solving equations. It allows us to reduce the complexity of mathematical expressions, making them easier to work with and understand. Mastering this concept is a significant step in developing algebraic proficiency.

When simplifying expressions, it's easy to make mistakes if we're not careful. Here are some common mistakes to watch out for:

  • Forgetting to distribute the negative sign: This is a very common mistake when removing parentheses preceded by a minus sign. Remember to distribute the negative sign to every term inside the parentheses. For example, in the expression 5 - (2x - 3), you need to distribute the negative sign to both 2x and -3, resulting in 5 - 2x + 3. If you forget to distribute the negative sign to the -3, you'll get the wrong answer.
  • Combining unlike terms: Only like terms can be combined. Don't try to add or subtract terms with different variables or exponents. For example, 3x and 2x^2 are not like terms and cannot be combined. Similarly, 4y and 5z are not like terms because they have different variables.
  • Incorrectly adding or subtracting coefficients: When combining like terms, make sure you add or subtract the coefficients correctly. For example, 7x - 3x should be 4x, not 10x or 4.
  • Not following the order of operations: Always follow the order of operations (PEMDAS) when simplifying expressions. This means dealing with parentheses first, then exponents, then multiplication and division, and finally addition and subtraction. Ignoring the order of operations can lead to incorrect results.
  • Dropping terms: Make sure you copy down all the terms correctly when rewriting the expression. It's easy to accidentally drop a term, especially if the expression is long. Always double-check your work to ensure that you haven't missed any terms.

By being aware of these common mistakes, you can significantly improve your accuracy when simplifying expressions.

Simplifying expressions by removing parentheses and combining like terms is a fundamental skill in algebra. By following the steps outlined in this article, you can effectively simplify expressions and make them easier to work with. Remember to distribute the negative sign correctly, combine only like terms, and follow the order of operations. With practice, you'll become proficient at simplifying expressions, which will be invaluable for solving equations and tackling more advanced mathematical concepts. Mastering this skill not only improves your ability to solve mathematical problems but also enhances your understanding of algebraic structures and manipulations. This understanding forms a solid foundation for further studies in mathematics and related fields. The ability to simplify expressions is a powerful tool that will serve you well throughout your mathematical journey.