Evaluating X³ - 3y When X = -3 And Y = 4 A Step-by-Step Guide

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In the realm of mathematics, algebraic expressions serve as the cornerstone for representing relationships between variables and constants. Evaluating these expressions involves substituting specific numerical values for the variables and performing the indicated operations. This process allows us to determine the numerical value of the expression for the given values of the variables. In this article, we will delve into the intricacies of evaluating algebraic expressions, providing a comprehensive guide to help you master this fundamental mathematical skill. Let's consider the expression x³ - 3y and evaluate it when x = -3 and y = 4. This example will serve as our guide as we explore the step-by-step process involved in evaluating algebraic expressions.

Step 1: Substitute the Given Values

The first step in evaluating an algebraic expression is to substitute the given numerical values for the corresponding variables. This involves replacing each variable in the expression with its assigned value. It's crucial to maintain the order of operations and pay close attention to signs (positive or negative) during this substitution process. For our expression, x³ - 3y, we are given that x = -3 and y = 4. Substituting these values into the expression, we get:

(-3)³ - 3(4)

This substitution step sets the stage for the subsequent calculations, allowing us to transform the algebraic expression into a numerical one.

Step 2: Apply the Order of Operations (PEMDAS/BODMAS)

Once the variables have been substituted with their respective values, the next crucial step is to apply the order of operations. This ensures that the calculations are performed in the correct sequence, leading to the accurate evaluation of the expression. The order of operations is commonly remembered by the acronyms PEMDAS or BODMAS:

  • Parentheses / Brackets
  • Exponents / Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Following this order, we first address the exponent in our expression, (-3)³. This means we need to multiply -3 by itself three times:

(-3)³ = (-3) × (-3) × (-3) = -27

Now our expression becomes:

-27 - 3(4)

Next, we perform the multiplication operation:

3(4) = 12

Our expression now simplifies to:

-27 - 12

Finally, we perform the subtraction operation:

-27 - 12 = -39

By diligently adhering to the order of operations, we have successfully simplified the expression and arrived at the correct numerical value.

Step 3: Simplify the Expression

After applying the order of operations, the next step is to simplify the expression as much as possible. This involves combining like terms, performing any remaining arithmetic operations, and reducing the expression to its simplest form. In our example, after performing the exponentiation and multiplication, we were left with:

-27 - 12

This expression can be simplified by performing the subtraction operation:

-27 - 12 = -39

Therefore, the simplified value of the expression x³ - 3y when x = -3 and y = 4 is -39. Simplifying the expression ensures that we arrive at the most concise and accurate numerical result.

Common Mistakes to Avoid

Evaluating algebraic expressions requires careful attention to detail, and certain common mistakes can lead to inaccurate results. Being aware of these pitfalls can help you avoid them and ensure the correctness of your evaluations. Here are some common mistakes to watch out for:

  1. Incorrectly Applying the Order of Operations: Failing to follow the order of operations (PEMDAS/BODMAS) can lead to errors in the calculation. Make sure to perform operations in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

  2. Sign Errors: Pay close attention to the signs (positive or negative) of the numbers and variables. A simple sign error can significantly alter the result. For example, subtracting a negative number is the same as adding its positive counterpart.

  3. Incorrect Substitution: Ensure that you substitute the values for the variables correctly. Double-check that you are replacing each variable with its assigned value and that you are not mixing up the values.

  4. Forgetting to Distribute: When dealing with expressions involving parentheses, remember to distribute any multiplication or division over the terms inside the parentheses. For example, a(b + c) = ab + ac.

  5. Combining Unlike Terms: Only like terms can be combined. Like terms are terms that have the same variables raised to the same powers. For example, 2x and 3x are like terms, but 2x and 3x² are not.

By being mindful of these common mistakes, you can improve your accuracy and confidence in evaluating algebraic expressions.

Examples and Practice Problems

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

Example 1:

Evaluate the expression 2x² + 5y - 3 when x = 2 and y = -1.

  • Step 1: Substitute the values:

    2(2)² + 5(-1) - 3

  • Step 2: Apply the order of operations:

    • Exponents: 2(4) + 5(-1) - 3
    • Multiplication: 8 - 5 - 3
    • Subtraction: 0

  • Solution: The value of the expression is 0.

Example 2:

Evaluate the expression (3a - 2b) / (a + b) when a = 4 and b = 2.

  • Step 1: Substitute the values:

    (3(4) - 2(2)) / (4 + 2)

  • Step 2: Apply the order of operations:

    • Parentheses (numerator): (12 - 4) / (4 + 2)
    • Parentheses (denominator): 8 / 6
    • Division: 4/3

  • Solution: The value of the expression is 4/3.

Practice Problems:

  1. Evaluate 4x - 7 when x = 5.
  2. Evaluate x² + 3x - 2 when x = -2.
  3. Evaluate (2a + b) / (a - b) when a = 3 and b = 1.

Working through these examples and practice problems will help you develop your skills in evaluating algebraic expressions and build confidence in your abilities.

Conclusion

Evaluating algebraic expressions is a fundamental skill in mathematics that forms the basis for more advanced concepts. By following the step-by-step guide outlined in this article, you can confidently and accurately evaluate a wide range of expressions. Remember to substitute the given values carefully, apply the order of operations diligently, and simplify the expression to its simplest form. By avoiding common mistakes and practicing regularly, you can master this essential mathematical skill and unlock new avenues of mathematical exploration. Evaluating the expression x³ - 3y when x = -3 and y = 4 resulted in -39, demonstrating the power and precision of algebraic evaluation. So, embrace the challenge, practice consistently, and watch your algebraic prowess flourish.