Evaluating The Algebraic Expression 2a²b³ - 3ab³x² - Bx Given A=2, B=-1, And X=1

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This article provides a comprehensive, step-by-step guide on evaluating the algebraic expression 2a²b³ - 3ab³x² - bx when given specific values for the variables a, b, and x. In this case, we are given a = 2, b = -1, and x = 1. This problem falls under the domain of basic algebra, specifically focusing on substitution and order of operations. We will break down the expression, substitute the values, and meticulously perform the calculations, emphasizing the importance of following the correct order of operations (PEMDAS/BODMAS) to arrive at the correct solution. This detailed explanation will be beneficial for students learning algebra, as well as anyone needing a refresher on algebraic evaluation.

Introduction to Algebraic Evaluation

In algebra, we often encounter expressions that contain variables. Variables are symbols (usually letters) that represent unknown values. To evaluate an algebraic expression, we replace the variables with their given numerical values and then simplify the expression using the order of operations. The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed. The commonly used acronyms for remembering this order are PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Both acronyms represent the same order:

  1. Parentheses/Brackets: Operations within parentheses or brackets are performed first.
  2. Exponents/Orders: Exponents (powers) and roots are evaluated next.
  3. Multiplication and Division: Multiplication and division are performed from left to right.
  4. Addition and Subtraction: Addition and subtraction are performed from left to right.

Understanding and adhering to the order of operations is crucial for obtaining accurate results when evaluating algebraic expressions. A slight deviation from this order can lead to a completely different answer. In the following sections, we will apply these principles to evaluate the expression at hand.

Step-by-Step Evaluation of 2a²b³ - 3ab³x² - bx

1. Understanding the Expression

The algebraic expression we are tasked with evaluating is 2a²b³ - 3ab³x² - bx. This expression consists of three terms, each involving variables a, b, and x raised to various powers and multiplied by coefficients. Let's break down each term:

  • 2a²b³: This term involves the product of the constant 2, the variable a squared (a²), and the variable b cubed (b³).
  • -3ab³x²: This term is the product of the constant -3, the variable a, the variable b cubed (b³), and the variable x squared (x²).
  • -bx: This term is the product of the negative of the variable b and the variable x.

Understanding the structure of the expression is the first step towards accurately evaluating it. We recognize the exponents, the coefficients, and the variables involved. This understanding will guide us in substituting the values and simplifying the expression according to the order of operations.

2. Substituting the Given Values

Now we substitute the given values for the variables into the expression. We are given a = 2, b = -1, and x = 1. Replacing the variables with their corresponding values, we get:

2(2)²(-1)³ - 3(2)(-1)³(1)² - (-1)(1)

This step is crucial, as it transforms the algebraic expression from a symbolic form to a numerical one. We have replaced the variables with their specific values, preparing us for the arithmetic calculations that will follow. It is important to be meticulous during substitution to avoid errors that could propagate through the rest of the solution.

3. Applying the Order of Operations (PEMDAS/BODMAS)

Now, we must simplify the expression obtained after substitution using the order of operations (PEMDAS/BODMAS). This means we will first address exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

Step 1: Evaluate Exponents

We have three exponents to evaluate:

  • (2)² = 2 * 2 = 4
  • (-1)³ = -1 * -1 * -1 = -1
  • (1)² = 1 * 1 = 1

Substituting these values back into the expression, we get:

2(4)(-1) - 3(2)(-1)(1) - (-1)(1)

Step 2: Perform Multiplication

Now we perform the multiplication operations from left to right:

  • 2(4)(-1) = 8(-1) = -8
  • 3(2)(-1)(1) = 6(-1)(1) = -6(1) = -6
  • (-1)(1) = -1

Substituting these products back into the expression, we get:

-8 - (-6) - (-1)

Step 3: Perform Subtraction

Finally, we perform the subtraction operations from left to right. Remember that subtracting a negative number is the same as adding its positive counterpart:

  • -8 - (-6) = -8 + 6 = -2
  • -2 - (-1) = -2 + 1 = -1

Therefore, the final result of the evaluation is -1.

4. Detailed Calculation Breakdown

To further clarify the process, let's recap the calculation with a detailed breakdown:

  1. Original Expression: 2a²b³ - 3ab³x² - bx
  2. Substitute Values: 2(2)²(-1)³ - 3(2)(-1)³(1)² - (-1)(1)
  3. Evaluate Exponents: 2(4)(-1) - 3(2)(-1)(1) - (-1)(1)
  4. Multiply First Term: 2(4)(-1) = -8
  5. Multiply Second Term: -3(2)(-1)(1) = -3(-2)(1) = 6
  6. Multiply Third Term: -(-1)(1) = 1
  7. Rewrite with Products: -8 + 6 + 1
  8. Add and Subtract: -8 + 6 = -2, then -2 + 1 = -1

This step-by-step breakdown provides a clear and concise explanation of each operation performed, ensuring a thorough understanding of the solution process.

Common Mistakes to Avoid

When evaluating algebraic expressions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them. Here are some of the most frequent errors:

  1. Incorrect Order of Operations: Failing to follow PEMDAS/BODMAS is a primary source of errors. Students may perform addition or subtraction before multiplication or division, or they may not evaluate exponents before other operations. Always double-check the order in which you are performing operations.

  2. Sign Errors: Mistakes involving negative signs are very common. For example, forgetting to square the negative sign when raising a negative number to an even power, or incorrectly distributing a negative sign across multiple terms. Pay close attention to signs and use parentheses when necessary to keep track of them.

  3. Substitution Errors: Errors can occur during the substitution process if values are assigned to the wrong variables or if the values are written incorrectly. Double-check that you have substituted the correct value for each variable.

  4. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect multiplication or addition, can also lead to wrong answers. Take your time and double-check your calculations.

  5. Forgetting the Distributive Property: When an expression involves parentheses, the distributive property must be applied correctly. Forgetting to distribute a term across all terms inside the parentheses can lead to significant errors.

By being mindful of these common mistakes and carefully reviewing your work, you can significantly improve your accuracy in evaluating algebraic expressions.

Practice Problems

To solidify your understanding of evaluating algebraic expressions, let's work through some practice problems.

Problem 1: Evaluate 3x² - 2y + z given x = -2, y = 3, and z = 1.

Solution:

  1. Substitute values: 3(-2)² - 2(3) + 1
  2. Evaluate exponents: 3(4) - 2(3) + 1
  3. Multiply: 12 - 6 + 1
  4. Add and Subtract: 12 - 6 = 6, then 6 + 1 = 7

Answer: 7

Problem 2: Evaluate -a³ + 4b² - 5c given a = 1, b = -1, and c = 2.

Solution:

  1. Substitute values: -(1)³ + 4(-1)² - 5(2)
  2. Evaluate exponents: -1 + 4(1) - 5(2)
  3. Multiply: -1 + 4 - 10
  4. Add and Subtract: -1 + 4 = 3, then 3 - 10 = -7

Answer: -7

Problem 3: Evaluate (x + y)² - 3z given x = 2, y = -1, and z = 3.

Solution:

  1. Substitute values: (2 + (-1))² - 3(3)
  2. Evaluate parentheses: (1)² - 3(3)
  3. Evaluate exponents: 1 - 3(3)
  4. Multiply: 1 - 9
  5. Subtract: -8

Answer: -8

These practice problems illustrate the application of the order of operations and the importance of careful substitution. Working through similar problems will help you build confidence and accuracy in evaluating algebraic expressions.

Conclusion

In this article, we have thoroughly explored the process of evaluating algebraic expressions by substituting given values for variables. We have emphasized the critical role of the order of operations (PEMDAS/BODMAS) in obtaining correct results. We have also highlighted common mistakes to avoid and provided practice problems to reinforce the concepts discussed. Mastering the ability to evaluate algebraic expressions is a fundamental skill in algebra and is essential for success in more advanced mathematical topics. By understanding the steps involved and practicing regularly, you can confidently tackle these types of problems.

Remember, the key to success in algebraic evaluation is:

  • Understanding the expression: Identify the variables, coefficients, and operations involved.
  • Careful substitution: Ensure that the correct values are substituted for the corresponding variables.
  • Strict adherence to the order of operations: Follow PEMDAS/BODMAS to perform operations in the correct sequence.
  • Practice, practice, practice: Work through a variety of problems to build your skills and confidence.

With these principles in mind, you are well-equipped to evaluate algebraic expressions accurately and efficiently.