Evaluating X³ - 2y + 3 Given X Equals -5 And Y Equals -2

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Introduction

In mathematics, evaluating expressions is a fundamental skill. This article focuses on evaluating the algebraic expression x³ - 2y + 3 when given specific values for the variables x and y. Specifically, we will substitute x = -5 and y = -2 into the expression and simplify to find the numerical value. This type of problem is crucial for understanding algebraic manipulation and order of operations. Let's embark on this mathematical journey together, ensuring that each step is clear and concise, ultimately arriving at the correct solution. This process underscores the importance of precision and attention to detail in mathematics, skills that are invaluable in various fields beyond academia.

Understanding the Expression

The expression x³ - 2y + 3 is an algebraic expression involving two variables, x and y. It consists of several terms: x³, -2y, and 3. The term x³ represents x raised to the power of 3, which means x multiplied by itself three times (x * x * x). The term -2y represents -2 multiplied by y. The term 3 is a constant. To evaluate this expression, we need to substitute the given values for x and y and follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding the structure of the expression is paramount before we delve into the substitution and simplification process. This foundation ensures that we approach the problem systematically and minimize the chances of errors. The interplay between variables, constants, and operations defines the essence of algebraic expressions, making it a cornerstone of mathematical understanding. We will explore each component of the expression in detail, ensuring a solid grasp before proceeding further.

Substituting the Values

The given values are x = -5 and y = -2. Substituting these values into the expression x³ - 2y + 3, we get: (-5)³ - 2(-2) + 3. This step is crucial as it transforms the algebraic expression into a numerical expression, which we can then simplify using the order of operations. The substitution process involves replacing each variable with its corresponding numerical value, being mindful of signs and parentheses. Parentheses are particularly important when dealing with negative numbers, as they ensure that the correct operations are performed. For instance, (-5)³ indicates that -5 is cubed, while -5³ would imply the negation of 5 cubed. Accuracy in substitution is paramount, as any error at this stage will propagate through the rest of the calculation. Therefore, meticulous attention to detail is required to avoid mistakes and ensure a correct final answer. This substitution lays the groundwork for the subsequent simplification, highlighting the interconnectedness of each step in the mathematical process.

Applying the Order of Operations (PEMDAS/BODMAS)

To simplify the expression (-5)³ - 2(-2) + 3, we must follow 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).

  1. Exponents: First, we evaluate (-5)³, which is (-5) * (-5) * (-5) = -125. This step is crucial as exponentiation takes precedence over multiplication and addition. Understanding the behavior of negative numbers raised to powers is essential here. An odd power of a negative number results in a negative number, while an even power results in a positive number.
  2. Multiplication: Next, we perform the multiplication: -2(-2) = 4. Multiplying two negative numbers results in a positive number.
  3. Addition and Subtraction: Now we have the expression -125 + 4 + 3. We perform the addition and subtraction from left to right: -125 + 4 = -121, and then -121 + 3 = -118.

Therefore, the simplified expression is -118. This systematic application of the order of operations ensures that we arrive at the correct result. Ignoring the order of operations can lead to incorrect answers, emphasizing the importance of adhering to these mathematical conventions.

Step-by-Step Calculation

Let's break down the calculation step by step to ensure clarity:

  1. Substitute the values: x³ - 2y + 3 becomes (-5)³ - 2(-2) + 3.
  2. Evaluate the exponent: (-5)³ = -125. The expression now becomes -125 - 2(-2) + 3.
  3. Perform the multiplication: -2(-2) = 4. The expression now becomes -125 + 4 + 3.
  4. Perform the addition and subtraction from left to right:
    • -125 + 4 = -121
    • -121 + 3 = -118

Therefore, the final result is -118. This step-by-step approach highlights the logical progression of the calculation, making it easier to follow and understand. Each step builds upon the previous one, leading to the ultimate solution. This meticulous breakdown minimizes the chances of errors and reinforces the importance of a systematic approach in mathematics. By visualizing each operation individually, we gain a deeper understanding of the overall process.

The Final Answer

After substituting the given values and simplifying the expression, we have determined that the value of x³ - 2y + 3 when x = -5 and y = -2 is -118. This final answer represents the culmination of our step-by-step calculations, highlighting the accuracy and precision we employed throughout the process. The negative value signifies that the combined effect of the terms in the expression results in a value less than zero. This result underscores the importance of understanding the properties of numbers and operations, as well as the significance of adhering to the order of operations. The answer * -118* is not just a numerical result; it's a testament to the logical and systematic approach we've taken to solve this problem. This outcome reinforces the power of mathematical reasoning and its ability to provide definitive solutions.

Conclusion

In conclusion, we have successfully evaluated the expression x³ - 2y + 3 for x = -5 and y = -2. The process involved substituting the given values, applying the order of operations (PEMDAS/BODMAS), and simplifying the resulting expression. By following a step-by-step approach, we arrived at the final answer of -118. This exercise demonstrates the importance of understanding algebraic expressions, the order of operations, and the substitution of values. It also highlights the need for accuracy and attention to detail in mathematical calculations. The ability to evaluate expressions is a fundamental skill in mathematics, and mastering this skill opens the door to more complex mathematical concepts and problem-solving techniques. This process serves as a building block for further mathematical exploration and application in various fields. The successful completion of this problem underscores the power of mathematical reasoning and its ability to provide clear and concise solutions.