Evaluate 3ab + 5b - 6 When A = -1 And B = 3

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Introduction

In the realm of mathematics, evaluating algebraic expressions is a fundamental skill. These expressions often involve variables, which are symbols that represent unknown values. To find the value of an expression, we substitute the given values for the variables and perform the indicated operations. This article delves into the process of evaluating a specific algebraic expression, 3ab + 5b - 6, when a = -1 and b = 3. We'll break down each step, providing a clear understanding of how to arrive at the solution. Whether you're a student grappling with algebra or simply seeking to refresh your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems.

Understanding the Expression

The expression we're dealing with is 3ab + 5b - 6. This is an algebraic expression because it combines numbers, variables (a and b), and mathematical operations (multiplication, addition, and subtraction). Let's break it down further:

  • 3ab: This term represents 3 multiplied by a and then multiplied by b. Remember that in algebra, when letters are written next to each other, it implies multiplication.
  • 5b: This term represents 5 multiplied by b.
  • -6: This is a constant term, meaning it's a fixed value that doesn't change.

To evaluate this expression, we need to substitute the given values for a and b and follow the order of operations (PEMDAS/BODMAS).

Step-by-Step Evaluation

Now, let's substitute the given values, a = -1 and b = 3, into the expression:

  1. Substitution: Replace 'a' with -1 and 'b' with 3 in the expression: 3(-1)(3) + 5(3) - 6

  2. Multiplication (3ab): Multiply 3, -1, and 3 together: 3 * -1 = -3 -3 * 3 = -9 So, the first term becomes -9.

  3. Multiplication (5b): Multiply 5 and 3 together: 5 * 3 = 15 So, the second term becomes 15.

  4. Rewriting the Expression: Now, substitute the results of the multiplications back into the expression: -9 + 15 - 6

  5. Addition and Subtraction (from left to right): First, add -9 and 15: -9 + 15 = 6 Then, subtract 6 from the result: 6 - 6 = 0

Therefore, the value of the expression 3ab + 5b - 6 when a = -1 and b = 3 is 0.

Importance of the Order of Operations

The 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), is crucial for evaluating expressions correctly. It dictates the sequence in which operations should be performed.

In our example, we first performed the multiplications (3ab and 5b) before carrying out the addition and subtraction. If we had ignored the order of operations and performed the addition before the multiplication, we would have arrived at an incorrect answer. This highlights the importance of adhering to PEMDAS/BODMAS to ensure accurate calculations.

Understanding and applying the order of operations is a cornerstone of algebraic proficiency. It allows us to simplify complex expressions systematically and arrive at the correct solution. Mastery of this concept is essential for success in algebra and beyond.

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 and improve your accuracy.

  • Incorrect Substitution: One frequent error is substituting the values for variables incorrectly. Double-check that you're replacing the correct variable with its corresponding value.
  • Ignoring the Order of Operations: As mentioned earlier, failing to follow PEMDAS/BODMAS is a significant source of errors. Always prioritize operations in the correct sequence.
  • Sign Errors: Dealing with negative numbers can be tricky. Pay close attention to the signs when multiplying, adding, or subtracting. A small sign error can significantly alter the outcome.
  • Arithmetic Mistakes: Even with the correct approach, simple arithmetic errors can occur. Take your time and double-check your calculations to minimize these mistakes.
  • Misinterpreting the Expression: Ensure you understand the expression correctly. For instance, 3ab means 3 multiplied by a and b, not 3 added to a and b.

By being mindful of these common errors and taking precautions to avoid them, you can enhance your ability to evaluate algebraic expressions accurately.

Real-World Applications

Evaluating algebraic expressions isn't just a theoretical exercise; it has numerous practical applications in various fields. From science and engineering to finance and computer programming, the ability to manipulate and solve expressions is essential.

  • Physics: Physicists use algebraic expressions to model physical phenomena, such as the motion of objects, the flow of electricity, and the behavior of waves. Evaluating these expressions allows them to make predictions and solve problems.
  • Engineering: Engineers rely on algebraic expressions to design structures, circuits, and systems. They use these expressions to calculate stress, strain, current, voltage, and other critical parameters.
  • Finance: Financial analysts use algebraic expressions to calculate interest rates, loan payments, investment returns, and other financial metrics. Evaluating these expressions helps them make informed decisions.
  • Computer Programming: Programmers use algebraic expressions to write algorithms and solve computational problems. Evaluating these expressions is crucial for ensuring that programs function correctly.

These are just a few examples of how evaluating algebraic expressions is used in the real world. The ability to work with expressions is a valuable skill that can open doors to a wide range of career paths.

Practice Problems

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

Problem 1: Evaluate 2x^2 - 3x + 1 when x = 4.

Solution:

  1. Substitute: 2(4)^2 - 3(4) + 1
  2. Exponents: 2(16) - 3(4) + 1
  3. Multiplication: 32 - 12 + 1
  4. Addition and Subtraction: 20 + 1 = 21

Therefore, the value of the expression is 21.

Problem 2: Evaluate (a + b)^2 - 4ab when a = 2 and b = -3.

Solution:

  1. Substitute: (2 + (-3))^2 - 4(2)(-3)
  2. Parentheses: (-1)^2 - 4(2)(-3)
  3. Exponents: 1 - 4(2)(-3)
  4. Multiplication: 1 - (-24)
  5. Subtraction: 1 + 24 = 25

Therefore, the value of the expression is 25.

By working through these practice problems, you can gain confidence in your ability to evaluate algebraic expressions. Remember to focus on understanding each step and applying the order of operations correctly.

Conclusion

In conclusion, evaluating algebraic expressions is a vital skill in mathematics and has numerous real-world applications. In this article, we explored the process of evaluating the expression 3ab + 5b - 6 when a = -1 and b = 3. We learned how to substitute values for variables, apply the order of operations, and avoid common mistakes. By understanding these concepts and practicing regularly, you can master the art of evaluating algebraic expressions and unlock new possibilities in mathematics and beyond. Remember, the key to success lies in a solid grasp of the fundamentals and consistent practice. So, keep exploring, keep learning, and keep challenging yourself!