Evaluating Algebraic Expressions A Step-by-Step Guide

Jul 10, 2025 by ADMIN 54 views

Evaluating the Expression (x^2 - xy + yz) Given x=3, y=-2, and z=-5

Introduction

In this article, we will delve into the process of evaluating the algebraic expression $(x^2 - xy + yz)$ given specific values for the variables $x$ , $y$ , and $z$ . The given values are $x = 3$ , $y = -2$ , and $z = -5$ . This type of problem is a fundamental concept in algebra, where the goal is to substitute the given values into the expression and simplify it to obtain a numerical result. Understanding how to evaluate expressions is crucial for solving more complex algebraic problems and is a cornerstone of mathematical proficiency. The steps involved are straightforward but require careful attention to detail, especially when dealing with negative numbers and exponents. We will break down the evaluation process step by step, ensuring a clear understanding of each operation. This exercise not only reinforces the order of operations (PEMDAS/BODMAS) but also enhances our ability to manipulate algebraic expressions with precision. By the end of this article, you will be confident in your ability to tackle similar algebraic evaluations and appreciate the importance of accuracy in mathematical calculations. So, let's embark on this mathematical journey and master the art of expression evaluation!

Step-by-Step Evaluation

To evaluate the expression $(x^2 - xy + yz)$ with $x = 3$ , $y = -2$ , and $z = -5$ , we will follow a systematic approach, substituting each variable with its corresponding value and then simplifying the expression according to the order of operations (PEMDAS/BODMAS). The order of operations dictates that we first handle any parentheses or brackets, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). This ensures that we arrive at the correct result by performing operations in the proper sequence. Let's begin by substituting the values into the expression. We replace $x$ with 3, $y$ with -2, and $z$ with -5, which gives us $(3^2 - (3)(-2) + (-2)(-5))$ . The next step is to address the exponents. We have $3^2$ , which means 3 multiplied by itself, resulting in 9. So, the expression now becomes $(9 - (3)(-2) + (-2)(-5))$ . Now, we focus on the multiplication operations. We have two instances of multiplication: $(3)(-2)$ and $(-2)(-5)$ . Multiplying 3 by -2 gives us -6, and multiplying -2 by -5 gives us 10. Remember, a negative number multiplied by a negative number results in a positive number. Substituting these products back into the expression, we have $(9 - (-6) + 10)$ . Next, we need to handle the subtraction of a negative number. Subtracting a negative number is the same as adding its positive counterpart. So, $(9 - (-6))$ becomes $(9 + 6)$ , which equals 15. The expression now simplifies to $(15 + 10)$ . Finally, we perform the addition. Adding 15 and 10 gives us 25. Therefore, the final result of evaluating the expression $(x^2 - xy + yz)$ with the given values is 25. This step-by-step evaluation highlights the importance of adhering to the order of operations and paying close attention to the signs of the numbers. Let's summarize these steps for clarity and reinforcement:

Substitute the values: Replace $x$ , $y$ , and $z$ with their given values.
Evaluate exponents: Calculate any exponential terms.
Perform multiplication: Multiply the terms as indicated.
Handle subtraction of negatives: Convert subtraction of a negative to addition of a positive.
Perform addition: Add the remaining terms.

By following these steps meticulously, you can confidently evaluate algebraic expressions and arrive at the correct answer every time.

Detailed Calculation

In this section, we will provide a detailed calculation of the expression $(x^2 - xy + yz)$ with $x = 3$ , $y = -2$ , and $z = -5$ , ensuring that each step is clearly explained. This meticulous approach aims to reinforce your understanding of the process and highlight the importance of precision in algebraic manipulations. We begin by substituting the given values into the expression, which yields:

$(3^2 - (3)(-2) + (-2)(-5))$

The first operation we address is the exponent. We have $3^2$ , which means 3 multiplied by itself:

$3^2 = 3 imes 3 = 9$

Now, we replace $3^2$ with 9 in the expression:

$(9 - (3)(-2) + (-2)(-5))$

Next, we focus on the multiplication operations. There are two multiplication operations to perform: $(3)(-2)$ and $(-2)(-5)$ . Let's calculate them one at a time.

$(3)(-2) = -6$

$(-2)(-5) = 10$

Remember that a positive number multiplied by a negative number results in a negative number, and a negative number multiplied by a negative number results in a positive number. Now, we substitute these products back into the expression:

$(9 - (-6) + 10)$

The next step involves dealing with the subtraction of a negative number. Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, $(9 - (-6))$ can be rewritten as:

$9 - (-6) = 9 + 6 = 15$

Substituting this result back into the expression, we have:

$(15 + 10)$

Finally, we perform the addition operation:

$15 + 10 = 25$

Thus, the final result of evaluating the expression $(x^2 - xy + yz)$ with $x = 3$ , $y = -2$ , and $z = -5$ is 25. This detailed calculation illustrates the step-by-step process and emphasizes the importance of accurate arithmetic and adherence to the order of operations. Each step is crucial in arriving at the correct answer, and neglecting any step or making a mistake in arithmetic can lead to an incorrect result. By meticulously following this process, you can enhance your algebraic skills and confidently solve similar problems.

Common Mistakes to Avoid

When evaluating algebraic expressions, it is crucial to be aware of common mistakes that can lead to incorrect answers. By understanding these pitfalls, you can develop strategies to avoid them and ensure accuracy in your calculations. One of the most frequent errors is the misapplication of the order of operations (PEMDAS/BODMAS). Forgetting to perform operations in the correct sequence can significantly alter the result. For example, failing to address exponents before multiplication or addition can lead to an incorrect answer. To avoid this, always double-check that you are following the correct order: Parentheses/Brackets, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Another common mistake involves incorrectly handling negative signs. Negative numbers can be tricky, especially when they are part of multiplication or subtraction. Remember that subtracting a negative number is the same as adding its positive counterpart, and multiplying two negative numbers results in a positive number. For example, $-( -5)$ is equal to $+5$ , and $(-2) imes (-3)$ is equal to $6$ . Failing to apply these rules correctly can lead to sign errors, which will affect the final result. Another potential source of error is arithmetic mistakes. Simple addition, subtraction, multiplication, or division errors can creep in, particularly when dealing with larger numbers or multiple operations. To minimize arithmetic errors, it is helpful to double-check your calculations, break down complex calculations into smaller steps, and use a calculator if necessary. Careless mistakes, such as transcribing numbers incorrectly or misreading the expression, can also occur. To avoid these, take your time, read the expression carefully, and double-check each step as you proceed. Writing each step clearly and neatly can also help you spot any errors more easily. Furthermore, forgetting to substitute values correctly is another common oversight. Make sure you replace each variable with its corresponding value accurately. It's a good practice to write out the substitution step clearly to avoid any confusion. Finally, skipping steps in the calculation process can increase the likelihood of making mistakes. While it might be tempting to perform multiple operations at once to save time, this can lead to errors. It's best to work through each step systematically, showing your work clearly. By being mindful of these common mistakes and taking the necessary precautions, you can significantly improve your accuracy when evaluating algebraic expressions. Let's summarize these common mistakes:

Misapplication of the order of operations
Incorrectly handling negative signs
Arithmetic mistakes
Careless mistakes
Forgetting to substitute values correctly
Skipping steps

Practice Problems

To solidify your understanding of evaluating algebraic expressions, it's essential to practice with a variety of problems. Practice not only reinforces the concepts but also helps you identify areas where you might need further clarification. In this section, we will present several practice problems that are similar to the example we discussed earlier. These problems will challenge you to apply the principles of substitution, order of operations, and careful arithmetic. Working through these exercises will boost your confidence and proficiency in algebraic evaluation. Here are some practice problems:

Evaluate the expression $(2x^2 + 3xy - y^2)$ if $x = 4$ and $y = -1$ .
Evaluate the expression $(a^3 - 2ab + bz)$ if $a = -2$ , $b = 3$ , and $z = 5$ .
Evaluate the expression $(5m - 2n^2 + mn)$ if $m = -3$ and $n = 2$ .
Evaluate the expression $(p^2 - 4pq + 3q^2)$ if $p = 1$ and $q = -4$ .
Evaluate the expression $(x^2yz - 2xyz^2)$ if $x = -1$ , $y = 2$ , and $z = -3$ .

For each problem, follow the step-by-step approach we outlined earlier:

Substitute the values of the variables into the expression.
Evaluate any exponents.
Perform multiplication and division from left to right.
Perform addition and subtraction from left to right.

Remember to pay close attention to negative signs and the order of operations to avoid common mistakes. After you have attempted each problem, compare your solutions with the answers provided below to check your work. If you encounter any discrepancies, carefully review your steps to identify where the error occurred. Working through these practice problems will not only enhance your understanding of algebraic evaluation but also improve your problem-solving skills in mathematics. Take your time, be meticulous, and enjoy the process of mastering these concepts. If you find yourself struggling with any particular problem, revisit the detailed explanation and examples discussed earlier in this article. Consistent practice is key to building confidence and competence in mathematics. So, let's dive into these problems and sharpen our algebraic skills!

Answers:

Conclusion

In this comprehensive guide, we have explored the process of evaluating algebraic expressions, focusing on the specific example of $(x^2 - xy + yz)$ given $x = 3$ , $y = -2$ , and $z = -5$ . We have dissected the problem step by step, emphasizing the importance of substitution, adhering to the order of operations (PEMDAS/BODMAS), and meticulous attention to detail, especially when dealing with negative signs and arithmetic operations. We began with an introduction that highlighted the significance of expression evaluation in algebra and its role as a fundamental skill in mathematics. We then proceeded to a detailed step-by-step evaluation, breaking down the process into manageable steps, from substituting the values to performing the final addition. A detailed calculation section provided a thorough walkthrough of each arithmetic operation, reinforcing the importance of accuracy and methodical problem-solving. We also addressed common mistakes to avoid, such as misapplying the order of operations, mishandling negative signs, and making arithmetic errors. By being aware of these pitfalls, you can develop strategies to prevent them and enhance your accuracy. Furthermore, we included a set of practice problems to solidify your understanding and provide you with opportunities to apply what you have learned. These problems are designed to challenge you and boost your confidence in tackling similar algebraic evaluations. Each problem is accompanied by an answer, allowing you to check your work and identify areas for improvement. The journey through this article has equipped you with the knowledge and skills necessary to evaluate algebraic expressions effectively. Whether you are a student learning algebra for the first time or someone looking to refresh your mathematical abilities, the principles and techniques discussed here will serve you well. Remember, practice is key to mastery, so continue to work through problems and challenge yourself. With consistent effort and attention to detail, you can confidently approach any algebraic evaluation and achieve accurate results. As you continue your mathematical journey, you will encounter more complex expressions and equations, but the foundational skills you have developed here will remain invaluable. So, embrace the challenge, stay focused, and enjoy the process of learning and growing in mathematics.