Calculate The Value Of Y = 6x + 8 When X Is -6

In this comprehensive guide, we will delve into the process of calculating the value of the algebraic expression y = 6x + 8 when x = -6. This is a fundamental concept in algebra, and understanding how to solve such problems is crucial for building a strong foundation in mathematics. We will break down the steps involved in a clear and concise manner, ensuring that you grasp the underlying principles and can confidently tackle similar problems in the future. This detailed explanation aims to not only provide the solution but also to enhance your understanding of variable substitution and algebraic evaluation. By the end of this guide, you will be well-equipped to solve a variety of algebraic expressions by substituting given values for variables. This skill is essential for further studies in mathematics and related fields, as it forms the basis for more complex problem-solving techniques. Let's embark on this mathematical journey together and unravel the intricacies of algebraic evaluation, ensuring that each step is clear, understandable, and contributes to your overall mathematical proficiency.

The equation we are working with is y = 6x + 8. This is a linear equation, which means that when plotted on a graph, it will form a straight line. The equation represents a relationship between two variables, y and x. In this case, y is the dependent variable, meaning its value depends on the value of x, which is the independent variable. The number 6 is the coefficient of x, indicating that x is multiplied by 6. The number 8 is a constant term, meaning it does not change regardless of the value of x. To calculate the value of y, we need to substitute the given value of x into the equation and perform the necessary arithmetic operations. This process of substitution is a cornerstone of algebra and allows us to determine the value of one variable when we know the value of another. Understanding the structure of the equation and the roles of the variables and constants is crucial for successfully evaluating algebraic expressions. Each part of the equation plays a specific role, and recognizing these roles helps in applying the correct procedures for solving the equation. The ability to identify the coefficient, variable, and constant terms is a fundamental skill in algebra and will serve you well in more advanced mathematical concepts. Therefore, mastering the interpretation of equations is as important as the calculation itself.

Our next step is to substitute the given value of x, which is -6, into the equation y = 6x + 8. This means we will replace the variable x with the number -6 in the equation. The equation then becomes y = 6(-6) + 8. It is important to use parentheses around the -6 to clearly indicate that we are multiplying 6 by -6 and not subtracting 6. This substitution is the key step in solving the equation, as it allows us to transform an equation with two variables into an equation with only one unknown, y. The accuracy of the substitution is crucial, as any mistake in this step will lead to an incorrect final answer. Therefore, double-checking the substitution is a good practice to ensure that the value of x has been correctly placed in the equation. This process of variable substitution is a fundamental technique in algebra and is used extensively in solving various types of equations and problems. By mastering this skill, you will be able to simplify complex equations and find solutions effectively. This step-by-step approach ensures that the solution is both accurate and easily understandable.

Now that we have substituted the value of x, we need to perform the multiplication operation. In the equation y = 6(-6) + 8, we have 6 multiplied by -6. When multiplying a positive number by a negative number, the result is always a negative number. Therefore, 6 multiplied by -6 equals -36. Our equation now looks like this: y = -36 + 8. It is crucial to follow the order of operations (PEMDAS/BODMAS), which dictates that multiplication should be performed before addition. Ignoring this rule would lead to an incorrect result. The multiplication step simplifies the equation further, bringing us closer to finding the value of y. This step highlights the importance of understanding the rules of multiplication with positive and negative numbers. A solid grasp of these rules is essential for accurate algebraic manipulation. The ability to correctly perform multiplication with negative numbers is a key skill in algebra and will prevent errors in more complex calculations. Therefore, this step reinforces the importance of precision and attention to detail in mathematical operations.

The final step in calculating the value of y is to perform the addition operation. We have the equation y = -36 + 8. Adding a positive number to a negative number is the same as subtracting the absolute value of the smaller number from the absolute value of the larger number and keeping the sign of the larger number. In this case, we have -36 and 8. The absolute value of -36 is 36, and the absolute value of 8 is 8. Subtracting 8 from 36 gives us 28. Since -36 has a larger absolute value and is negative, the result will be negative. Therefore, -36 + 8 = -28. This means that the value of y is -28. This final step demonstrates the application of addition rules with negative numbers, which is a fundamental concept in arithmetic and algebra. Understanding how to add numbers with different signs is crucial for accurate calculations in various mathematical contexts. The ability to correctly perform addition with negative numbers is a key skill that prevents errors and ensures the accuracy of the final result. This step concludes the process of evaluating the algebraic expression, and the result highlights the importance of following the correct order of operations and applying the rules of arithmetic.

In conclusion, we have successfully calculated the value of y in the equation y = 6x + 8 when x = -6. By following the steps of substitution, multiplication, and addition, we found that y = -28. This process demonstrates the importance of understanding algebraic equations and the rules of arithmetic. The ability to substitute values into equations and perform the necessary operations is a fundamental skill in mathematics. Mastering these skills will enable you to solve a wide range of algebraic problems and build a solid foundation for more advanced mathematical concepts. The step-by-step approach we followed ensures clarity and accuracy in the solution process. This example serves as a valuable exercise in algebraic evaluation and reinforces the importance of precision and attention to detail in mathematical calculations. By understanding each step and the underlying principles, you can confidently apply these techniques to solve similar problems in the future. This comprehensive guide has equipped you with the knowledge and skills necessary to tackle algebraic expressions and find solutions effectively. Remember, practice is key to mastering mathematical concepts, so continue to practice and explore different types of equations to enhance your problem-solving abilities.