Evaluating 2x³y - 3x²y²z + 2xy³z For X=-2, Y=1, Z=0 A Step-by-Step Guide

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This article delves into the process of evaluating a given algebraic expression. Specifically, we will focus on the expression 2x³y - 3x²y²z + 2xy³z, where the variables x, y, and z are assigned the values -2, 1, and 0 respectively. We will break down each step of the calculation, ensuring a clear understanding of the order of operations and the application of exponent rules. This exercise is fundamental in algebra, as it reinforces the concept of variable substitution and expression simplification. Understanding how to correctly evaluate expressions is crucial for solving more complex equations and problems in mathematics and related fields.

Understanding the Expression

The algebraic expression we are working with is 2x³y - 3x²y²z + 2xy³z. This expression consists of three terms separated by subtraction and addition operations. Each term involves variables x, y, and z raised to different powers and multiplied by constant coefficients. To evaluate this expression, we need to substitute the given values for x, y, and z and then perform the arithmetic operations according to the order of operations (PEMDAS/BODMAS). Let's first recall the order of operations:

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

This order is essential to ensure we arrive at the correct result. In our expression, we have exponents, multiplication, subtraction, and addition. We will first address the exponents, then the multiplications, and finally the subtractions and additions. This systematic approach is crucial for accurate evaluation. Before diving into the calculations, it's helpful to rewrite the expression with the substituted values. This helps in visualizing the steps and reducing errors. This step involves replacing every instance of x with -2, y with 1, and z with 0.

Step-by-Step Evaluation

Now, let's proceed with the step-by-step evaluation of the expression 2x³y - 3x²y²z + 2xy³z when x = -2, y = 1, and z = 0. First, we substitute the given values into the expression:

2(-2)³(1) - 3(-2)²(1)²(0) + 2(-2)(1)³(0)

Next, we evaluate the exponents. Remember that (-2)³ means -2 multiplied by itself three times, and (-2)² means -2 multiplied by itself twice:

  • (-2)³ = -2 * -2 * -2 = -8
  • (-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(-8)(1) - 3(4)(1)(0) + 2(-2)(1)(0)

Now, we perform the multiplications from left to right:

  • 2(-8)(1) = -16
  • 3(4)(1)(0) = 0
  • 2(-2)(1)(0) = 0

Substituting these results back into the expression, we have:

-16 - 0 + 0

Finally, we perform the subtractions and additions:

-16 - 0 + 0 = -16

Therefore, the value of the expression 2x³y - 3x²y²z + 2xy³z when x = -2, y = 1, and z = 0 is -16. This detailed step-by-step approach highlights the importance of following the order of operations to achieve the correct result. Errors in evaluating exponents or performing multiplication before addition/subtraction can lead to an incorrect answer. This careful methodology is essential for success in algebra and beyond. In the next section, we'll explore the impact of the z = 0 on simplifying the expression.

The Impact of z = 0

In the previous section, we meticulously evaluated the expression 2x³y - 3x²y²z + 2xy³z for the given values of x, y, and z. A keen observation reveals that the terms involving z become zero due to multiplication by zero. This is a crucial point in understanding algebraic simplification. When any term is multiplied by zero, the entire term becomes zero. In our case, the second term (-3x²y²z) and the third term (2xy³z) both contain z, which is equal to 0. Therefore, these terms effectively vanish from the expression when we substitute z = 0. This simplifies the expression considerably.

Let's revisit the expression:

2x³y - 3x²y²z + 2xy³z

When z = 0, the expression becomes:

2x³y - 3x²y²(0) + 2xy³(0)

This simplifies to:

2x³y - 0 + 0

Which further reduces to:

2x³y

This simplified expression, 2x³y, makes the evaluation process significantly easier. We only need to substitute the values of x and y into this simplified form. This demonstrates the power of simplification in algebra. Recognizing and utilizing simplifications can save time and reduce the chances of making errors in calculations. The effect of z being zero highlights the importance of identifying such opportunities for simplification before diving into complex calculations. Understanding the properties of zero in multiplication is a fundamental concept in mathematics. This principle allows us to eliminate entire terms from an expression, making the evaluation process more manageable. This strategic approach to simplification is a valuable tool for solving mathematical problems efficiently.

Evaluating the Simplified Expression

Having simplified the expression to 2x³y when z = 0, we can now evaluate it with x = -2 and y = 1. This simplified form makes the calculation much more straightforward. The process involves substituting the values of x and y into the expression and then performing the necessary arithmetic operations. Let's substitute the values:

2(-2)³(1)

First, we evaluate the exponent:

(-2)³ = -2 * -2 * -2 = -8

Substituting this back into the expression, we get:

2(-8)(1)

Now, we perform the multiplication:

2(-8)(1) = -16

Therefore, the value of the simplified expression 2x³y when x = -2 and y = 1 is -16. This result matches the value we obtained when evaluating the original, more complex expression. This reinforces the correctness of our simplification and evaluation steps. The ease with which we arrived at the answer using the simplified expression highlights the benefits of identifying and utilizing opportunities for simplification. This skill is particularly valuable in more complex algebraic problems, where simplification can significantly reduce the computational burden. Furthermore, evaluating the simplified expression serves as a verification step. By comparing the result with the evaluation of the original expression, we can confirm that no errors were made during the simplification process. This practice of verification is crucial for ensuring the accuracy of mathematical solutions.

Conclusion

In this article, we have thoroughly explored the evaluation of the algebraic expression 2x³y - 3x²y²z + 2xy³z when x = -2, y = 1, and z = 0. We began by understanding the expression and the order of operations, then proceeded with a step-by-step evaluation, carefully calculating each term. We also highlighted the significant impact of z = 0, which allowed us to simplify the expression to 2x³y. This simplification not only made the evaluation process easier but also demonstrated the power of algebraic manipulation in problem-solving. The final evaluation of the simplified expression yielded the same result as the original expression, confirming the accuracy of our work. This exercise reinforces several important concepts in algebra:

  • Variable Substitution: The ability to correctly substitute values for variables is fundamental to evaluating expressions and solving equations.
  • Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) is crucial for obtaining the correct result.
  • Exponent Rules: Understanding and applying exponent rules is essential for simplifying and evaluating expressions involving powers.
  • Simplification: Identifying opportunities for simplification, such as the impact of multiplication by zero, can significantly reduce the complexity of a problem.
  • Verification: Verifying the solution by different methods, such as evaluating both the original and simplified expressions, ensures accuracy.

By mastering these concepts, one can confidently tackle more complex algebraic problems. The ability to evaluate expressions accurately and efficiently is a cornerstone of mathematical proficiency, paving the way for success in advanced topics such as calculus and linear algebra. This exploration serves as a valuable reminder of the importance of careful, methodical work and the power of simplification in mathematics. The skills and understanding gained from this exercise are transferable to a wide range of mathematical and scientific applications. The process of evaluating algebraic expressions is not just a mathematical exercise; it is a fundamental skill that underpins many areas of science, engineering, and technology. Understanding how to manipulate and evaluate expressions is essential for anyone seeking to solve real-world problems using mathematical tools.