Isolating Y² In 9x² + 7y² = 42 A Step By Step Guide

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In the realm of mathematics, the ability to isolate a specific variable within an equation is a fundamental skill. This process allows us to express that variable in terms of other variables and constants, providing valuable insights into the relationship between them. In this comprehensive guide, we will delve into the steps involved in isolating in the equation 9x² + 7y² = 42. Our primary focus is to not just arrive at the correct answer but also to provide a thorough, step-by-step explanation that enhances understanding. This is crucial for students and anyone looking to brush up on their algebra skills. Understanding how to manipulate equations is essential for solving more complex problems in mathematics and related fields. We will break down each step, ensuring clarity and precision, and also highlight common pitfalls to avoid. By the end of this guide, you will be equipped with the knowledge to confidently tackle similar problems and gain a deeper appreciation for algebraic manipulations. Let's embark on this mathematical journey together and unravel the solution for . By carefully following each step and understanding the underlying principles, you'll strengthen your algebraic foundation and improve your problem-solving skills.

Step-by-Step Solution: Isolating y²

When it comes to isolating in the equation 9x² + 7y² = 42, it's crucial to follow a systematic approach. The equation we're working with is a linear equation in terms of , which makes the process straightforward if we adhere to the correct algebraic principles. The primary goal here is to manipulate the equation so that is alone on one side, and all other terms are on the other side. This involves a series of algebraic steps, each building upon the previous one, ensuring we maintain the equation's balance. First, we'll address the term that does not involve , which is 9x². To isolate , we need to eliminate this term from the left-hand side of the equation. This is achieved by subtracting 9x² from both sides of the equation, a fundamental operation that preserves the equality. After this subtraction, the equation will be in a form where the term is closer to being isolated. Next, we'll deal with the coefficient of , which is 7. To completely isolate , we need to eliminate this coefficient. This is done by dividing both sides of the equation by 7. This step ensures that stands alone on one side of the equation, giving us the desired expression for in terms of and constants. Each of these steps is rooted in basic algebraic principles, and understanding them is key to mastering equation manipulation. By carefully applying these steps, we can confidently isolate and express it in its simplest form.

1. Subtract 9x² from Both Sides

The initial step in isolating involves addressing the term that doesn't contain , which is 9x². To do this, we apply the fundamental algebraic principle of maintaining equality by performing the same operation on both sides of the equation. In this case, we subtract 9x² from both the left-hand side (LHS) and the right-hand side (RHS) of the equation. This subtraction is a pivotal move because it begins the process of isolating the term we're interested in, 7y². By subtracting 9x² from both sides, we effectively remove it from the left side, bringing us one step closer to having alone. This operation is based on the additive property of equality, which states that if you subtract the same quantity from both sides of an equation, the equality remains true. This is a cornerstone of algebraic manipulation and is crucial for solving equations. It's important to perform this step carefully, ensuring that the subtraction is applied correctly to both sides to avoid any errors in the subsequent steps. The resulting equation after this subtraction sets the stage for the next operation, where we will deal with the coefficient of . This step-by-step approach ensures that we methodically isolate while preserving the integrity of the equation. Understanding why and how this step works is essential for mastering algebraic problem-solving.

2. Divide Both Sides by 7

Following the subtraction of 9x² from both sides, the next crucial step in isolating is to address the coefficient of , which is 7. To completely isolate , we need to eliminate this coefficient. This is achieved by applying another fundamental algebraic principle: division. We divide both sides of the equation by 7, ensuring that the equality is maintained. This division is a pivotal operation because it directly targets the coefficient of , effectively undoing the multiplication by 7. This step is based on the multiplicative property of equality, which states that if you divide both sides of an equation by the same non-zero quantity, the equality remains true. This principle is fundamental to solving equations and isolating variables. It's important to note that we can only divide by a non-zero number; in this case, dividing by 7 is perfectly valid. The act of dividing both sides by 7 separates from its coefficient, leaving it isolated on one side of the equation. This is a significant milestone in our solution process, bringing us closer to the final answer. The result of this division provides us with the expression for in terms of and constants, which is the ultimate goal of isolating the variable. By carefully performing this division, we ensure the accuracy of our solution and demonstrate a clear understanding of algebraic principles.

Final Result: The Isolated Form of y²

After performing the necessary algebraic manipulations, the final result of isolating in the equation 9x² + 7y² = 42 is a concise and informative expression. This expression reveals the direct relationship between and , providing valuable insight into the equation's properties. The isolated form of not only gives us the solution but also allows for further analysis and applications. For instance, we can easily substitute different values of x into the equation to find corresponding values of y, which is particularly useful in graphing and understanding the behavior of the equation. The final isolated form of is a testament to the power of algebraic manipulation, showcasing how we can transform an equation to highlight specific relationships between variables. This result is not just an answer; it's a tool for further exploration and understanding of the mathematical relationship defined by the original equation. It is the culmination of a step-by-step process, each step building upon the previous one to achieve the desired outcome. This final form is the most simplified representation of in terms of , and it is the key to unlocking further insights into the equation's nature and solutions. Understanding how to arrive at this result is crucial for anyone seeking to master algebraic problem-solving.

Conclusion: Mastering the Art of Isolating Variables

In conclusion, the process of isolating in the equation 9x² + 7y² = 42 demonstrates the fundamental principles of algebraic manipulation. By following a systematic approach, we were able to successfully isolate and express it in terms of and constants. This skill is not only essential for solving equations but also for understanding the relationships between variables in various mathematical contexts. Mastering the art of isolating variables is a cornerstone of algebraic proficiency, enabling us to tackle more complex problems and gain deeper insights into mathematical concepts. Throughout this guide, we emphasized the importance of each step, from subtracting terms to dividing by coefficients, ensuring that each operation was performed with precision and accuracy. The final result, the isolated form of , is a testament to the power of algebraic manipulation and the clarity it brings to mathematical expressions. By understanding the underlying principles and applying them methodically, we can confidently isolate variables in a wide range of equations. This skill is not just for mathematicians; it's a valuable tool for anyone who uses mathematics in their daily lives, whether in science, engineering, finance, or any other field. The ability to manipulate equations and isolate variables is a key to unlocking the solutions to many real-world problems, making it an indispensable skill for lifelong learning and problem-solving.

The correct answer is B. y² = 6 - (9/7)x².