Simplify Algebraic Expressions Remove Grouping Symbols And Combine Like Terms

Simplifying algebraic expressions is a fundamental skill in mathematics. It allows you to take complex equations and reduce them to their most basic form, making them easier to understand and manipulate. In this comprehensive guide, we will walk you through the process of simplifying the expression $[10 - 8(x - 5y)] + 9(8x + y)$, breaking down each step and explaining the underlying principles. We will cover the necessary concepts such as the order of operations, the distributive property, combining like terms, and removing grouping symbols. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic simplification problems.

Understanding the Order of Operations

Before we dive into the specifics of our example, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order dictates the sequence in which we perform operations to ensure we arrive at the correct answer. In the realm of simplifying algebraic expressions, the order of operations is your roadmap to success. Think of PEMDAS/BODMAS as a set of instructions that guarantees consistency and accuracy in your calculations. It's like a universal language in mathematics, ensuring that everyone interprets an expression in the same way. For example, consider the expression $2 + 3 × 4$. If we were to add first, we would get $5 × 4 = 20$. But according to PEMDAS/BODMAS, we should multiply first: $3 × 4 = 12$, then add: $2 + 12 = 14$. This simple example highlights the importance of following the correct order. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are simply mnemonic devices to help you remember the correct sequence. Whether you use PEMDAS or BODMAS, the key is to apply the order consistently. In our journey to simplify algebraic expressions, we'll encounter parentheses, multiplication, addition, and subtraction. By adhering to PEMDAS/BODMAS, we'll navigate these operations smoothly and arrive at the simplified form of our expression. Remember, the goal of simplification is not just to get an answer, but to represent the expression in its most concise and understandable form. This makes it easier to work with in further calculations or when solving equations. Therefore, a solid grasp of the order of operations is not just a preliminary step, but a fundamental tool in your algebraic toolkit.

The Distributive Property

The distributive property is a key concept that allows us to multiply a single term by multiple terms within a set of parentheses. The distributive property is a cornerstone of algebra, allowing us to expand expressions and remove parentheses. It states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. In simpler terms, it means that you can multiply a single term by each term inside a set of parentheses and then add the results. This property is crucial for simplifying expressions that involve parentheses and is frequently used in combination with the order of operations. To truly grasp the power of the distributive property, it's helpful to visualize it geometrically. Imagine a rectangle with a width of 'a' and a length of 'b + c'. The total area of the rectangle is a(b + c). We can also divide this rectangle into two smaller rectangles, one with dimensions 'a' and 'b', and the other with dimensions 'a' and 'c'. The areas of these smaller rectangles are 'ab' and 'ac' respectively. The sum of these areas, 'ab + ac', is equal to the total area, demonstrating the distributive property visually. In our example expression, $[10 - 8(x - 5y)] + 9(8x + y)$, we have two instances where the distributive property is applicable: $-8(x - 5y)$ and $9(8x + y)$. Let's break down each one separately. First, we'll distribute the -8 across the terms inside the first parentheses: $-8 * x = -8x$ and $-8 * -5y = 40y$. Remember that multiplying two negative numbers results in a positive number. Next, we'll distribute the 9 across the terms inside the second parentheses: $9 * 8x = 72x$ and $9 * y = 9y$. By applying the distributive property in this way, we've successfully removed the parentheses and expanded the expression, paving the way for the next step in simplification. The distributive property is not limited to expressions with two terms inside the parentheses. It can be applied to any number of terms. For example, $a(b + c + d) = ab + ac + ad$. Furthermore, the distributive property works with subtraction as well: $a(b - c) = ab - ac$. The key is to multiply the term outside the parentheses by each term inside, paying close attention to the signs. Mastering the distributive property is essential for simplifying algebraic expressions, solving equations, and working with polynomials. It's a versatile tool that will serve you well throughout your mathematical journey. Practice applying the distributive property with different expressions to build your confidence and fluency.

Combining Like Terms

After applying the distributive property, you'll often find yourself with an expression containing multiple terms. The next step is to combine like terms to further simplify the expression. Combining like terms is a fundamental technique in algebra that allows you to consolidate terms with the same variable and exponent. This process involves identifying terms that share the same variable raised to the same power and then adding or subtracting their coefficients. A coefficient is the numerical part of a term. For example, in the term $5x$, 5 is the coefficient and $x$ is the variable. Terms are considered “like terms” if they have the exact same variable part. For instance, $3x$ and $-7x$ are like terms because they both have the variable $x$ raised to the power of 1. Similarly, $2y^2$ and $9y^2$ 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 the exponents are different. Neither are $5x$ and $5y$ because the variables are different. To combine like terms, you simply add or subtract their coefficients while keeping the variable part the same. For example, $3x + (-7x) = (3 - 7)x = -4x$. Similarly, $2y^2 + 9y^2 = (2 + 9)y^2 = 11y^2$. Constant terms (numbers without variables) are also considered like terms and can be combined. For example, $5 + 8 = 13$. In our example expression, after applying the distributive property, we'll have terms with $x$, terms with $y$, and constant terms. Our goal is to identify and combine these like terms to reduce the expression to its simplest form. This process makes the expression easier to understand and work with in subsequent calculations or when solving equations. Combining like terms is not just a matter of simplifying expressions. It's also about understanding the structure of algebraic expressions and recognizing the relationships between different terms. This skill is crucial for more advanced topics in algebra, such as solving equations, factoring polynomials, and working with rational expressions. By mastering the technique of combining like terms, you'll be able to manipulate algebraic expressions with confidence and ease. Practice identifying and combining like terms in various expressions to solidify your understanding. This will not only improve your algebraic skills but also enhance your overall mathematical problem-solving abilities.

Removing Grouping Symbols

Grouping symbols, such as parentheses, brackets, and braces, play a crucial role in dictating the order of operations. Removing grouping symbols is a vital step in simplifying algebraic expressions. These symbols indicate which operations should be performed first. Parentheses are the most common grouping symbols, followed by brackets and braces. In the order of operations (PEMDAS/BODMAS), operations within grouping symbols are always performed before operations outside of them. Therefore, when simplifying algebraic expressions, one of the first steps is often to remove these symbols. To effectively remove grouping symbols, you need to understand how they interact with the operations around them, particularly multiplication and the distributive property. If a term is multiplied by an expression inside a grouping symbol, you must distribute that term across all the terms within the grouping symbol. This is precisely what we discussed in the section on the distributive property. If there is a negative sign in front of a grouping symbol, it's equivalent to multiplying the expression inside by -1. This means you need to change the sign of every term inside the grouping symbol when you remove it. For example, $-(a + b)$ becomes $-a - b$, and $-(a - b)$ becomes $-a + b$. If there is a positive sign in front of a grouping symbol, you can simply remove the grouping symbol without changing any signs. For example, $+(a + b)$ is simply $a + b$. When dealing with nested grouping symbols (grouping symbols within grouping symbols), you should start by removing the innermost grouping symbols first and work your way outwards. This ensures that you apply the order of operations correctly. For example, in the expression $2[3 + (4 - 1)]$, you would first simplify the expression inside the parentheses $(4 - 1)$, then the expression inside the brackets $[3 + 3]$, and finally multiply by 2. In our example expression, $[10 - 8(x - 5y)] + 9(8x + y)$, we have two sets of parentheses. We've already addressed how to remove them using the distributive property. After distributing the -8 and the 9, the parentheses will be gone, and we can proceed to combine like terms. The key to successfully removing grouping symbols is to be methodical and pay close attention to signs. A small mistake in distributing a negative sign can lead to an incorrect final answer. Practice working with different types of grouping symbols and nested expressions to build your confidence. Mastering this skill is essential for simplifying complex algebraic expressions and solving equations.

Step-by-Step Solution

Now, let's apply these concepts to simplify the given expression step by step:

Original expression: $[10 - 8(x - 5y)] + 9(8x + y)$

1. Apply the distributive property to the first set of parentheses: $10 - 8x + 40y + 9(8x + y)$

2. Apply the distributive property to the second set of parentheses: $10 - 8x + 40y + 72x + 9y$

3. Combine like terms: Group the x terms, the y terms, and the constant terms: $(-8x + 72x) + (40y + 9y) + 10$

4. Add the coefficients of the x terms and the y terms: $64x + 49y + 10$

Therefore, the simplified expression is $64x + 49y + 10$.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct answer. One of the most frequent errors is incorrectly applying the distributive property. This often involves forgetting to distribute the term to all terms inside the parentheses or making mistakes with signs. For example, when distributing a negative sign, it's crucial to remember to change the sign of every term inside the parentheses. Another common mistake is failing to adhere to the order of operations. This can lead to performing addition or subtraction before multiplication or division, resulting in an incorrect simplification. Always remember PEMDAS/BODMAS and follow the correct sequence of operations. Combining unlike terms is another frequent error. Remember that you can only combine terms that have the same variable raised to the same power. For example, $3x$ and $3x^2$ are not like terms and cannot be combined. Confusing coefficients and exponents is also a common mistake. A coefficient is the numerical part of a term, while an exponent indicates the power to which a variable is raised. Be careful not to add coefficients when you should be adding exponents, or vice versa. Sign errors are also a significant source of mistakes in algebraic simplification. Pay close attention to positive and negative signs, especially when distributing negative signs or combining like terms. A small error with a sign can completely change the answer. Another pitfall is overlooking the simplification of constant terms. Make sure to combine all constant terms in the expression to arrive at the most simplified form. Finally, not double-checking your work can lead to overlooking simple errors. After simplifying an expression, take a few moments to review each step and ensure that you haven't made any mistakes. By being mindful of these common errors and taking steps to avoid them, you can significantly improve your accuracy in simplifying algebraic expressions. Practice is key to mastering these skills and developing a strong understanding of algebraic principles.

Practice Problems

To solidify your understanding, try simplifying the following expressions:

1. $3(2x - 5) + 4x$
2. $7[1 - (3y + 2)]$
3. $-2(4a - b) - 5b$

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics. By understanding the order of operations, the distributive property, combining like terms, and removing grouping symbols, you can confidently tackle complex expressions and reduce them to their simplest forms. Remember to practice regularly and be mindful of common mistakes to improve your accuracy and proficiency. With consistent effort, you'll master the art of simplifying algebraic expressions and unlock new possibilities in your mathematical journey.