How To Simplify The Expression 4(1 - 3x) + 7x - 8 A Step-by-Step Guide

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Simplifying algebraic expressions is a fundamental skill in mathematics. It allows us to rewrite complex expressions in a more manageable and understandable form. In this article, we will walk through the process of simplifying the expression 4(1 - 3x) + 7x - 8 step by step. We will cover the key concepts and techniques involved, providing a clear and comprehensive explanation that will help you master this essential skill. Understanding how to simplify expressions is crucial for solving equations, working with formulas, and tackling more advanced mathematical problems. This article aims to provide a thorough understanding of how to simplify such expressions, making it easier to tackle more complex algebraic problems in the future.

Understanding the Basics of Algebraic Expressions

Before diving into the simplification process, it's crucial to understand the basic components of algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. Mathematical operations include addition, subtraction, multiplication, and division. In our expression, 4(1 - 3x) + 7x - 8, x is the variable, and 4, 1, -3, 7, and -8 are constants. The operations involved are multiplication, subtraction, and addition. To effectively simplify algebraic expressions, you need to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which operations should be performed to ensure the correct result. Understanding like terms is also essential; like terms are terms that contain the same variable raised to the same power, allowing them to be combined through addition or subtraction. For instance, 7x and -12x are like terms, while 7x and 7x² are not. With a solid grasp of these basics, you’re well-equipped to tackle the simplification process.

Step-by-Step Simplification Process

The given expression is 4(1 - 3x) + 7x - 8. To simplify this, we will follow a series of steps, applying the order of operations and combining like terms. Each step is crucial for arriving at the simplified form. First, we need to address the parentheses. According to the order of operations (PEMDAS), we must handle any operations within parentheses before anything else. In this case, we have 4(1 - 3x). This means we need to distribute the 4 across both terms inside the parentheses. Distribution involves multiplying the term outside the parentheses by each term inside. So, we multiply 4 by 1 and 4 by -3x. This gives us 4 * 1 - 4 * 3x, which simplifies to 4 - 12x. The expression now becomes 4 - 12x + 7x - 8. Next, we identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, -12x and 7x are like terms because they both contain the variable x raised to the power of 1. Similarly, 4 and -8 are like terms because they are both constants. To combine like terms, we simply add or subtract their coefficients. The coefficient is the numerical part of the term. So, we combine -12x + 7x, which gives us -5x. We also combine 4 - 8, which gives us -4. Finally, we rewrite the expression with the combined terms. The simplified expression is -5x - 4. This is the simplest form of the original expression. By following these steps carefully, we have successfully simplified the expression by distributing, combining like terms, and adhering to the order of operations.

Step 1: Distribution

The first key step in simplifying the expression 4(1 - 3x) + 7x - 8 is distribution. Distribution is the process of multiplying a term outside parentheses by each term inside the parentheses. This step is crucial for eliminating the parentheses and allowing us to combine like terms later on. In our expression, we have 4(1 - 3x). The number 4 is outside the parentheses, and the terms inside are 1 and -3x. To distribute, we multiply 4 by each term inside the parentheses. First, we multiply 4 by 1, which gives us 4 * 1 = 4. Next, we multiply 4 by -3x, which gives us 4 * (-3x) = -12x. So, after distributing the 4, the expression 4(1 - 3x) becomes 4 - 12x. Now, we replace the original expression within the parentheses with this simplified form. The entire expression now reads 4 - 12x + 7x - 8. This step of distribution is essential because it transforms the expression into a form where we can identify and combine like terms. By correctly distributing, we ensure that we are following the order of operations (PEMDAS), which dictates that we must handle operations within parentheses before other operations. The result of this distribution, 4 - 12x, is a crucial intermediate step towards the final simplified expression. Understanding and correctly applying distribution is fundamental to simplifying algebraic expressions.

Step 2: Combining Like Terms

After the distribution step, our expression is now 4 - 12x + 7x - 8. The next critical step in simplifying this expression is combining like terms. Like terms are terms that contain the same variable raised to the same power. In this expression, we have two types of terms: terms with the variable x and constant terms (numbers without variables). The terms -12x and 7x are like terms because they both contain the variable x raised to the power of 1. The constant terms 4 and -8 are also like terms because they are both numerical values without variables. To combine like terms, we add or subtract their coefficients. The coefficient is the numerical part of the term. For the x terms, we have -12x + 7x. To combine these, we add their coefficients: -12 + 7 = -5. So, -12x + 7x simplifies to -5x. For the constant terms, we have 4 - 8. To combine these, we subtract 8 from 4, which gives us 4 - 8 = -4. Now that we have combined the like terms, we rewrite the expression using the simplified terms. The expression 4 - 12x + 7x - 8 becomes -5x - 4. This step is crucial because it reduces the number of terms in the expression, making it simpler and easier to understand. By correctly identifying and combining like terms, we are one step closer to the final simplified form. Combining like terms not only simplifies the expression but also makes it easier to work with in further mathematical operations.

Step 3: Final Simplified Expression

After distributing and combining like terms, we have arrived at the final simplified expression. From the previous steps, we distributed the 4 in 4(1 - 3x) to get 4 - 12x, and then combined like terms in 4 - 12x + 7x - 8. We combined -12x and 7x to get -5x, and we combined 4 and -8 to get -4. This leaves us with the expression -5x - 4. This expression is the simplest form of the original expression 4(1 - 3x) + 7x - 8. There are no more like terms to combine, and no further operations can be performed. The simplified expression is now in a clear and concise form, making it easier to understand and use in further calculations or problem-solving. The process of simplifying algebraic expressions is crucial in mathematics because it allows us to reduce complex expressions to their most basic form, which can help in solving equations, understanding mathematical relationships, and making calculations more straightforward. By following the steps of distribution and combining like terms, we can effectively simplify a wide range of algebraic expressions. The final simplified expression, -5x - 4, represents the original expression in its most manageable form, ready for use in any mathematical context.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's crucial to avoid common mistakes that can lead to incorrect results. One frequent error is failing to distribute correctly. For example, in the expression 4(1 - 3x) + 7x - 8, some might mistakenly multiply 4 only by 1, forgetting to multiply it by -3x. This would lead to an incorrect expression. Another common mistake is combining unlike terms. Remember, you can only combine terms that have the same variable raised to the same power. For instance, adding -12x and 4 is incorrect because they are not like terms. Similarly, forgetting the order of operations (PEMDAS) can lead to errors. Always perform operations within parentheses first, then exponents, then multiplication and division, and finally addition and subtraction. A mistake in the order can completely change the result. Sign errors are also prevalent, especially when dealing with negative numbers. For example, when combining like terms like 4 - 8, it's easy to mistakenly calculate it as 4. Always double-check your signs to ensure accuracy. Another pitfall is not simplifying completely. Make sure you have combined all like terms and performed all possible operations before considering the expression fully simplified. To avoid these mistakes, practice consistently and double-check each step. Pay close attention to detail, especially when distributing and combining like terms. Understanding the rules and practicing regularly will help you simplify expressions accurately and efficiently. By being mindful of these common errors, you can improve your algebraic skills and achieve the correct solutions.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, working through practice problems is essential. Practice helps reinforce the concepts and techniques discussed, making the process more intuitive. Here are a few practice problems for you to try:

  1. Simplify: 2(3x + 4) - 5x + 1
  2. Simplify: -3(2 - x) + 4x - 7
  3. Simplify: 5(x - 2) + 3(2x + 1)
  4. Simplify: 6 - 2(4x - 3) + 5x
  5. Simplify: -4(1 + x) - 2(3 - x)

For each problem, remember to follow the steps we discussed: first, distribute any terms outside parentheses, then combine like terms. Take your time, and double-check your work to avoid common mistakes. After you've attempted these problems, you can check your answers. The solutions are provided below:

  1. Solution: x + 9
  2. Solution: 7x - 13
  3. Solution: 11x - 7
  4. Solution: -3x + 12
  5. Solution: -2x - 10

Working through these problems will help you become more confident and proficient in simplifying algebraic expressions. If you encounter any difficulties, review the steps and explanations provided in this article. Consistent practice is the key to mastering this fundamental skill in mathematics. By tackling a variety of problems, you'll develop a deeper understanding of the process and improve your ability to simplify expressions accurately and efficiently.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, allowing us to rewrite complex expressions in a more manageable form. We walked through the process of simplifying the expression 4(1 - 3x) + 7x - 8 step by step, covering key concepts such as distribution and combining like terms. We also emphasized the importance of following the order of operations (PEMDAS) and avoiding common mistakes. By understanding these principles and practicing consistently, you can confidently simplify various algebraic expressions. Mastering this skill is not only essential for success in algebra but also for tackling more advanced mathematical problems in the future. Remember to distribute correctly, combine like terms accurately, and double-check your work to avoid errors. Practice regularly with a variety of problems to reinforce your understanding and improve your proficiency. Simplifying expressions is a fundamental building block for further mathematical studies, and with a solid grasp of these concepts, you'll be well-prepared to tackle more complex challenges. Keep practicing, and you'll find that simplifying algebraic expressions becomes second nature, empowering you to solve a wide range of mathematical problems with ease and confidence.