Evaluating The Algebraic Expression 3ab + 5b - 6 When A = -1 And B = 3

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In this article, we will delve into the process of evaluating an algebraic expression given specific values for its variables. The expression we will be working with is 3ab + 5b - 6, and we are tasked with finding its value when a = -1 and b = 3. 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 algebra and beyond. We will break down the process step-by-step, ensuring clarity and comprehension for readers of all backgrounds. This involves first substituting the given values into the expression, then following the order of operations (PEMDAS/BODMAS) to simplify and arrive at the final answer. The article will not only provide the solution but also emphasize the importance of each step, helping readers develop a solid understanding of algebraic evaluation. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this comprehensive guide will equip you with the knowledge and confidence to tackle similar problems.

Step-by-Step Solution

To evaluate the expression 3ab + 5b - 6 when a = -1 and b = 3, we will follow these steps:

  1. Substitution: Replace the variables a and b with their given values in the expression. This gives us: 3(-1)(3) + 5(3) - 6.
  2. Multiplication: Perform the multiplication operations. Remember that multiplying a positive number by a negative number results in a negative number. So, 3 * (-1) * 3 = -9, and 5 * 3 = 15. Our expression now looks like this: -9 + 15 - 6.
  3. Addition and Subtraction: Perform the addition and subtraction operations from left to right. -9 + 15 = 6, and then 6 - 6 = 0. Therefore, the value of the expression is 0.

This step-by-step approach highlights the importance of following the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). By adhering to this order, we ensure that we arrive at the correct answer. In this case, the expression involves multiplication, addition, and subtraction. We first handled the multiplication, then proceeded with addition and subtraction from left to right. This methodical approach minimizes the risk of errors and provides a clear pathway to the solution. Furthermore, understanding the rules of integer arithmetic, such as multiplying positive and negative numbers, is crucial for accurate calculations. The final result, 0, demonstrates a specific instance where the combination of terms in the expression cancels out, showcasing the dynamic nature of algebraic expressions.

Importance of Order of Operations (PEMDAS/BODMAS)

The order of operations, often remembered by the acronyms PEMDAS or BODMAS, is a fundamental principle in mathematics. It dictates the sequence in which mathematical operations should be performed to ensure consistent and accurate results. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). BODMAS stands for Brackets, Orders (powers and square roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). Both acronyms represent the same order of operations. Failing to adhere to this order can lead to incorrect answers, especially in more complex expressions involving multiple operations. In the example expression 3ab + 5b - 6, we first performed the multiplication (3ab and 5b) before proceeding with the addition and subtraction. This is because multiplication and division have a higher precedence than addition and subtraction according to PEMDAS/BODMAS. If we were to perform the addition before the multiplication, the result would be significantly different. For instance, if we incorrectly added 5b to -6 before multiplying, we would be deviating from the established mathematical convention, leading to a wrong solution. The order of operations ensures that mathematical expressions are evaluated consistently, regardless of who is solving them. This consistency is essential for clear communication and accurate problem-solving in mathematics and related fields. Mastering PEMDAS/BODMAS is therefore a crucial skill for anyone working with mathematical expressions.

Common Mistakes to Avoid

When evaluating algebraic expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy. One of the most frequent errors is disregarding the order of operations (PEMDAS/BODMAS). As discussed earlier, performing operations in the wrong order can lead to an incorrect result. Another common mistake is incorrectly handling negative signs. For example, when substituting a negative value for a variable, it's crucial to pay close attention to the signs in the expression. In our example, substituting a = -1 requires careful attention to the multiplication of 3, -1, and b. A forgotten negative sign can completely change the outcome. Another error arises from incorrectly distributing numbers or combining unlike terms. In more complex expressions, proper distribution is necessary before simplification. Students may also mistakenly combine terms that are not like terms (e.g., adding a term with a variable to a constant term). Furthermore, arithmetic errors during multiplication, division, addition, or subtraction are also common sources of mistakes. It's essential to double-check calculations to ensure accuracy. A simple arithmetic error can propagate through the entire problem, leading to a wrong final answer. Lastly, careless transcription of the problem can also cause errors. Ensure you accurately copy the expression and the values of the variables before starting the evaluation. By being mindful of these common mistakes and practicing careful problem-solving techniques, you can significantly reduce the likelihood of errors and improve your algebraic skills. Regular practice and attention to detail are key to mastering algebraic evaluation.

Practice Problems

To solidify your understanding of evaluating algebraic expressions, working through practice problems is essential. Here are a few problems similar to the one we solved, which you can try on your own:

  1. Evaluate the expression 2x + 7y - 4 when x = 2 and y = -1.
  2. Find the value of 4p - 3q + 9 when p = -3 and q = 5.
  3. What is the value of -5m + 2n - 10 when m = 4 and n = -2?
  4. Calculate the value of 6a - 8b + 12 when a = -1 and b = -3.
  5. Evaluate the expression -3c + 9d - 7 when c = 2 and d = 1.

These practice problems cover a range of scenarios, including positive and negative values, and different combinations of operations. To solve these problems effectively, follow the same step-by-step approach we used in the example. First, substitute the given values for the variables. Then, perform the multiplication operations. Finally, carry out the addition and subtraction operations from left to right, adhering to the order of operations (PEMDAS/BODMAS). After solving each problem, check your answer carefully. If you encounter any difficulties, review the steps outlined in this article and try to identify where you might have made an error. Working through these practice problems will not only reinforce your understanding of evaluating algebraic expressions but also build your confidence in tackling similar problems in the future. Remember, practice is key to mastering any mathematical concept, and algebraic evaluation is no exception. Regular practice will help you develop fluency and accuracy in solving these types of problems.

Conclusion

In conclusion, evaluating algebraic expressions is a fundamental skill in mathematics. In this article, we have demonstrated a step-by-step approach to evaluating the expression 3ab + 5b - 6 when a = -1 and b = 3. The process involves substituting the given values for the variables, performing the multiplication operations, and then carrying out the addition and subtraction operations from left to right, adhering to the order of operations (PEMDAS/BODMAS). We have also emphasized the importance of understanding and applying the order of operations correctly, as well as avoiding common mistakes such as incorrect handling of negative signs and arithmetic errors. By following the steps outlined in this article and practicing regularly, you can develop confidence and proficiency in evaluating algebraic expressions. Remember to pay close attention to detail, double-check your calculations, and be mindful of the order of operations. Mastering this skill will not only help you succeed in algebra but also provide a solid foundation for more advanced mathematical concepts. The practice problems provided in this article offer an opportunity to further solidify your understanding and hone your skills. Embrace the challenge of solving these problems and use them as a tool to reinforce your learning. Algebraic evaluation is a crucial building block for mathematical success, and with consistent effort and practice, you can master this skill and excel in your mathematical journey. The value of the expression 3ab + 5b - 6 when a = -1 and b = 3 is 0.