Evaluate The Expression (2a - 3b) / 5c Given A = 2, B = -2, And C = 1.

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In this article, we will delve into the process of evaluating algebraic expressions by substituting given values for variables. Specifically, we will focus on the expression (2a - 3b) / 5c, where we are provided with the values a = 2, b = -2, and c = 1. This type of problem is fundamental in algebra and serves as a building block for more complex mathematical concepts. Understanding how to correctly substitute values and simplify expressions is crucial for success in higher-level mathematics. We'll walk through the steps involved, providing clear explanations and highlighting common pitfalls to avoid. By the end of this discussion, you'll have a solid understanding of how to tackle similar problems with confidence. The ability to evaluate expressions accurately is not just important for academic purposes; it also has practical applications in various fields, such as engineering, physics, and computer science, where formulas and equations are used extensively. Therefore, mastering this skill is a valuable investment in your mathematical journey. We will explore the order of operations, which is paramount in achieving the correct result. We'll also emphasize the importance of paying close attention to signs, especially when dealing with negative numbers. Furthermore, we'll discuss how to break down the problem into smaller, manageable steps to minimize the chances of errors. This methodical approach will not only help you solve this specific problem but also equip you with a general strategy for tackling any algebraic expression evaluation. Let's embark on this mathematical exploration together and unlock the secrets of expression evaluation!

Step-by-Step Solution

To evaluate the expression (2a - 3b) / 5c, given that a = 2, b = -2, and c = 1, we need to follow these steps:

  1. Substitute the given values for the variables in the expression. This means replacing 'a' with 2, 'b' with -2, and 'c' with 1. The expression becomes: (2 * 2 - 3 * (-2)) / (5 * 1)

    Substitution is a fundamental concept in algebra, where we replace variables with their numerical values to obtain a numerical result. It's crucial to perform the substitution accurately, ensuring that each variable is replaced with its corresponding value. In this case, we've carefully replaced 'a' with 2, 'b' with -2, and 'c' with 1. This substitution sets the stage for the next steps, where we'll simplify the expression using the order of operations. A common mistake is to substitute the values incorrectly, which can lead to a wrong answer. Therefore, double-checking the substitution is always a good practice. The act of substitution transforms the algebraic expression into a numerical expression, which we can then evaluate using arithmetic operations. This process bridges the gap between abstract algebraic symbols and concrete numerical values, allowing us to find the value of the expression for the given conditions. Mastering substitution is essential for solving a wide range of algebraic problems, including equations, inequalities, and functions. It's a cornerstone of algebraic manipulation and a skill that will serve you well in your mathematical journey. Understanding the concept of substitution also helps in comprehending the relationship between variables and their values, which is crucial for modeling real-world scenarios using mathematical equations. By practicing substitution, you'll develop a deeper understanding of algebraic expressions and their behavior. This understanding will empower you to tackle more complex problems with confidence and precision.

  2. Simplify the numerator. We need to follow the order of operations (PEMDAS/BODMAS), which means performing multiplication before subtraction. First, we multiply 2 * 2, which equals 4. Then, we multiply 3 * (-2), which equals -6. The numerator now looks like this: 4 - (-6)

    Simplifying the numerator is a crucial step in evaluating the expression. This involves performing the arithmetic operations in the correct order, according to the order of operations (PEMDAS/BODMAS). In this case, we have both multiplication and subtraction in the numerator. According to the order of operations, we must perform the multiplication operations before the subtraction. We first multiply 2 by 2, which gives us 4. Then, we multiply 3 by -2, which gives us -6. It's important to pay close attention to the signs, especially when dealing with negative numbers. Multiplying a positive number by a negative number results in a negative number. Now, our numerator has been simplified to 4 - (-6). This is where many students make mistakes by not correctly handling the subtraction of a negative number. Subtracting a negative number is equivalent to adding the positive of that number. Therefore, 4 - (-6) is the same as 4 + 6. The careful application of the order of operations and the correct handling of negative numbers are essential for simplifying the numerator accurately. This step lays the foundation for the next steps, where we will further simplify the expression and ultimately arrive at the final answer. Understanding the order of operations is not just important for this specific problem but also for any mathematical expression that involves multiple operations. It ensures that we perform the calculations in the correct sequence, leading to the correct result. Mastering this concept is fundamental to success in algebra and beyond.

  3. Simplify further. Subtracting a negative number is the same as adding its positive counterpart. So, 4 - (-6) becomes 4 + 6, which equals 10. The numerator is now simplified to 10.

    Further simplification of the numerator is the next logical step in our evaluation process. As we established in the previous step, subtracting a negative number is equivalent to adding its positive counterpart. Therefore, the expression 4 - (-6) can be rewritten as 4 + 6. This transformation is crucial for accurate calculation and is a common rule in arithmetic that students should master. Once we rewrite the expression, the addition becomes straightforward. Adding 4 and 6 gives us 10. So, the numerator is now completely simplified to 10. This simplification is a significant milestone in our evaluation process. We've successfully reduced the numerator to a single numerical value, making the subsequent steps easier to handle. The ability to simplify expressions by applying basic arithmetic rules is a fundamental skill in mathematics. It allows us to transform complex expressions into simpler forms, making them easier to understand and evaluate. This skill is not only essential for solving algebraic problems but also for various other mathematical applications. By mastering the art of simplification, you'll be able to tackle a wide range of mathematical challenges with confidence. This step-by-step approach to simplification is a key strategy for avoiding errors and ensuring accuracy. Each step builds upon the previous one, leading us closer to the final solution. By carefully simplifying each part of the expression, we minimize the chances of making mistakes and ensure that our answer is correct.

  4. Simplify the denominator. We multiply 5 * 1, which equals 5. The denominator is now simplified to 5.

    Simplifying the denominator is an equally important step in evaluating the expression. In this case, the denominator is 5c, and we know that c = 1. So, we need to multiply 5 by 1. This multiplication is straightforward and results in 5. Therefore, the denominator is simplified to 5. Simplifying the denominator is crucial because it allows us to express the entire expression as a single fraction. This makes it easier to perform the final division and obtain the final result. A simplified denominator also provides a clearer picture of the expression's value and its relationship to the numerator. In some cases, the denominator may involve more complex operations, such as addition, subtraction, or even exponents. In such cases, we would need to apply the order of operations to simplify the denominator correctly. However, in this specific problem, the denominator simplification is relatively simple, involving only a single multiplication. The key takeaway here is that simplifying both the numerator and the denominator is essential for evaluating any fractional expression. This ensures that we are working with the simplest possible form of the expression, minimizing the chances of errors and making the calculations more manageable. The ability to simplify denominators is a valuable skill in algebra and is essential for solving equations, inequalities, and other mathematical problems. By mastering this skill, you'll be able to confidently tackle a wide range of mathematical challenges involving fractions and rational expressions. This step-by-step approach to simplification, applied to both the numerator and the denominator, is a cornerstone of algebraic manipulation.

  5. Divide the simplified numerator by the simplified denominator. We now have 10 / 5, which equals 2.

    Dividing the simplified numerator by the simplified denominator is the final step in evaluating the expression. We have successfully simplified the numerator to 10 and the denominator to 5. Now, we simply need to perform the division operation: 10 / 5. This division is a basic arithmetic operation that yields the result 2. Therefore, the value of the expression (2a - 3b) / 5c, when a = 2, b = -2, and c = 1, is 2. This final step brings together all the previous simplifications and calculations, culminating in the final answer. It demonstrates the power of step-by-step problem-solving, where complex expressions are broken down into smaller, manageable parts. The division operation is the inverse of multiplication and is a fundamental arithmetic operation. In this case, it allows us to express the ratio between the numerator and the denominator as a single numerical value. Understanding division is crucial for various mathematical applications, including fractions, ratios, and proportions. The result of the division, 2, represents the final value of the expression for the given values of the variables. This value is a single number that encapsulates the entire expression's behavior under the specified conditions. The ability to perform this final division accurately is a testament to the careful and methodical approach we've taken throughout the evaluation process. This step-by-step approach not only ensures the correct answer but also provides a clear and logical pathway for understanding the problem and its solution. By mastering this process, you'll be well-equipped to tackle similar problems with confidence and precision.

Final Answer

Therefore, 2a−3b5c=2\frac{2a - 3b}{5c} = 2 when a = 2, b = -2, and c = 1.

Key Takeaways

  • Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) to ensure accurate calculations.
  • Sign Conventions: Pay close attention to signs, especially when dealing with negative numbers.
  • Step-by-Step Approach: Break down complex expressions into smaller, manageable steps to minimize errors.
  • Substitution: Substitute values carefully and double-check your work.

Common Mistakes to Avoid

  • Incorrect Substitution: Ensure that you substitute the correct values for the variables.
  • Ignoring Order of Operations: Failing to follow the order of operations can lead to incorrect results.
  • Sign Errors: Be extra cautious when dealing with negative numbers and their operations.

Practice Problems

To solidify your understanding, try evaluating the following expressions using the same principles:

  1. (4x + 2y) / z, where x = 3, y = -1, z = 2
  2. (a^2 - b^2) / c, where a = 5, b = 3, c = 4
  3. (2m - n) / (p + 1), where m = 4, n = -2, p = 1

By working through these practice problems, you'll reinforce your understanding of expression evaluation and develop your problem-solving skills. Remember to focus on each step, pay attention to detail, and double-check your work. The more you practice, the more confident and proficient you'll become in tackling algebraic expressions. These skills are essential for success in mathematics and have wide-ranging applications in various fields. So, embrace the challenge, and enjoy the journey of mathematical discovery!