Evaluating -4y - C When C = 11/14 And Y = -2/7 In Simplest Form

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In mathematics, evaluating expressions is a fundamental skill. It involves substituting given values for variables and performing the indicated operations to find the numerical result. This article focuses on the process of evaluating the algebraic expression −4y−c-4y - c when the variables cc and yy are assigned specific fractional values. We will explore the step-by-step substitution, arithmetic operations with fractions, and simplification techniques to arrive at the final answer in its simplest form. A solid understanding of these concepts is crucial for success in algebra and higher-level mathematics, as it forms the basis for solving equations, analyzing functions, and modeling real-world scenarios. By mastering the process of evaluating expressions, students can develop a deeper appreciation for the symbolic language of mathematics and its power to represent and solve a wide range of problems. This article aims to provide a clear and comprehensive guide to evaluating the given expression, reinforcing the core principles of algebraic manipulation and fractional arithmetic. We will begin by defining the expression and the given values, then proceed with the substitution and simplification steps. Along the way, we will emphasize the importance of accuracy and attention to detail in mathematical calculations, as even small errors can lead to incorrect results. By working through this example, readers will gain confidence in their ability to evaluate similar expressions and apply these skills to more complex problems in the future.

Before we begin the evaluation process, it's crucial to have a clear understanding of the expression and the given values. The expression we need to evaluate is −4y−c-4y - c. This expression involves two variables, yy and cc, each representing a numerical value. The values assigned to these variables are: c=1114c = \frac{11}{14} and y=−27y = -\frac{2}{7}. These values are fractions, which means we'll need to apply the rules of fractional arithmetic when performing the calculations. The expression −4y−c-4y - c indicates that we need to multiply the value of yy by -4, and then subtract the value of cc from the result. It's important to pay close attention to the signs of the numbers, as a mistake in the sign can change the entire outcome. Understanding the order of operations (PEMDAS/BODMAS) is also critical here. Multiplication must be performed before subtraction. By carefully dissecting the expression and noting the given values, we set the stage for a successful evaluation. This initial step ensures that we have a clear roadmap for the calculations ahead, minimizing the chances of errors and maximizing the efficiency of the process. In the following sections, we will delve into the step-by-step substitution and simplification of this expression, building upon this foundational understanding. Remember, a strong grasp of the basics is the key to tackling more complex mathematical problems with confidence and accuracy. Therefore, ensure you fully understand the expression and the given values before moving on to the next step. This will lay a solid groundwork for your understanding and simplify the subsequent calculations.

Now, let's proceed with the step-by-step evaluation of the expression −4y−c-4y - c when c=1114c = \frac{11}{14} and y=−27y = -\frac{2}{7}.

  1. Substitute the values: Replace the variables yy and cc in the expression with their given values. −4y−c=−4(−27)−1114-4y - c = -4(-\frac{2}{7}) - \frac{11}{14}

  2. Perform the multiplication: Multiply −4-4 by −27-\frac{2}{7}. Remember that multiplying two negative numbers results in a positive number. −4(−27)=−4×−27=87-4(-\frac{2}{7}) = \frac{-4 \times -2}{7} = \frac{8}{7}

    Now the expression becomes: 87−1114\frac{8}{7} - \frac{11}{14}

  3. Find a common denominator: To subtract fractions, they must have a common denominator. The least common multiple (LCM) of 7 and 14 is 14. Convert 87\frac{8}{7} to an equivalent fraction with a denominator of 14. 87=8×27×2=1614\frac{8}{7} = \frac{8 \times 2}{7 \times 2} = \frac{16}{14}

    Now the expression is: 1614−1114\frac{16}{14} - \frac{11}{14}

  4. Subtract the fractions: Subtract the numerators while keeping the common denominator. 1614−1114=16−1114=514\frac{16}{14} - \frac{11}{14} = \frac{16 - 11}{14} = \frac{5}{14}

The result of evaluating the expression −4y−c-4y - c when c=1114c = \frac{11}{14} and y=−27y = -\frac{2}{7} is 514\frac{5}{14}. To ensure that this is the simplest form, we need to check if the numerator and denominator have any common factors other than 1. In this case, the numerator is 5, which is a prime number, and the denominator is 14, which has factors of 2 and 7. Since 5 and 14 have no common factors other than 1, the fraction 514\frac{5}{14} is indeed in its simplest form. Therefore, the final answer is 514\frac{5}{14}. This comprehensive step-by-step evaluation demonstrates the process of substituting values, performing arithmetic operations with fractions, and simplifying the result to its lowest terms. By following this methodical approach, you can confidently evaluate various algebraic expressions and ensure accurate and simplified answers. Remember to always double-check your calculations and look for opportunities to simplify fractions to their simplest forms. This practice will enhance your understanding of mathematical concepts and improve your problem-solving skills. The ability to evaluate expressions accurately is a cornerstone of algebra and essential for tackling more advanced mathematical challenges. This example serves as a valuable guide to mastering this skill and building a strong foundation in mathematics.

In conclusion, we have successfully evaluated the expression −4y−c-4y - c when c=1114c = \frac{11}{14} and y=−27y = -\frac{2}{7}. By carefully substituting the given values, performing the necessary arithmetic operations, and simplifying the result, we arrived at the final answer of 514\frac{5}{14}. This process highlights the importance of several key concepts in mathematics, including the order of operations, fractional arithmetic, and simplification techniques. The ability to evaluate expressions accurately is a fundamental skill that is essential for success in algebra and beyond. It forms the basis for solving equations, analyzing functions, and modeling real-world scenarios. By mastering this skill, students can develop a deeper understanding of mathematical concepts and enhance their problem-solving abilities. The step-by-step approach demonstrated in this article provides a clear and concise method for evaluating similar expressions. By following this approach, students can avoid common errors and ensure accurate results. Furthermore, the emphasis on simplification underscores the importance of presenting answers in their simplest form, which is a crucial aspect of mathematical communication. As you continue your journey in mathematics, remember that practice is key to mastering any skill. Work through various examples, pay attention to details, and don't hesitate to seek help when needed. With consistent effort and a solid understanding of the fundamentals, you can confidently tackle more complex mathematical challenges and achieve your academic goals. The knowledge and skills gained through this exercise will serve as a valuable foundation for future mathematical endeavors and empower you to approach problems with greater confidence and competence.