Isolating Y² In The Equation 4x² + 25y² = 100 A Step-by-Step Solution

In the realm of mathematics, manipulating equations to isolate specific variables is a fundamental skill. This article delves into the process of isolating y² in the equation 4x² + 25y² = 100, providing a step-by-step guide and exploring the underlying concepts. We will dissect the equation, apply algebraic principles, and arrive at the correct solution, while also highlighting the significance of this process in various mathematical contexts. Whether you're a student grappling with algebraic manipulations or simply seeking to enhance your mathematical understanding, this comprehensive guide will equip you with the knowledge and confidence to tackle similar problems.

Demystifying the Equation: 4x² + 25y² = 100

The equation 4x² + 25y² = 100 represents an ellipse, a fundamental shape in conic sections. Understanding the properties of ellipses and how their equations are structured is crucial for solving problems related to them. The equation is in the standard form of an ellipse centered at the origin, where the coefficients of x² and y² determine the shape and orientation of the ellipse. Isolating y² in this equation allows us to express y in terms of x, which is essential for graphing the ellipse, finding its intercepts, and analyzing its behavior. This process not only provides a solution to the specific problem but also reinforces the broader concept of manipulating equations to gain insights into the relationships between variables.

The Importance of Isolating Variables

Isolating a variable in an equation is a cornerstone of algebraic manipulation. It allows us to express one variable in terms of others, revealing the relationships between them. In the case of y², isolating it in the equation 4x² + 25y² = 100 provides a direct expression for y² as a function of *x². This is crucial for various mathematical operations, such as graphing the equation, finding the values of y for specific values of x, and analyzing the behavior of the equation. Moreover, isolating variables is a fundamental step in solving systems of equations, where we aim to find the values of multiple variables that satisfy a set of equations simultaneously. The ability to isolate variables is a testament to one's understanding of algebraic principles and their application in solving mathematical problems.

Step-by-Step Guide to Isolating y²

To isolate y² in the equation 4x² + 25y² = 100, we will follow a systematic approach using basic algebraic operations. Each step is carefully explained to ensure clarity and understanding.

Step 1: Isolate the term containing y²

Our initial goal is to isolate the term 25y² on one side of the equation. To achieve this, we subtract 4x² from both sides of the equation:

4x² + 25y² - 4x² = 100 - 4x²

This simplifies to:

25y² = 100 - 4x²

This step utilizes the fundamental property of equality, which states that performing the same operation on both sides of an equation maintains the equality. By subtracting 4x² from both sides, we effectively move it from the left side to the right side, isolating the y² term.

Step 2: Divide both sides by the coefficient of y²

Now that we have 25y² isolated, we need to isolate y² itself. To do this, we divide both sides of the equation by the coefficient of y², which is 25:

(25y²) / 25 = (100 - 4x²) / 25

This simplifies to:

y² = (100 - 4x²) / 25

This step again relies on the property of equality. By dividing both sides by 25, we isolate y² on the left side. However, the right side can be further simplified.

Step 3: Simplify the expression

The expression on the right side can be simplified by dividing each term in the numerator by the denominator:

y² = 100/25 - (4x²)/25

This simplifies to:

y² = 4 - (4/25)x²

This is the final form of the equation, where y² is isolated and expressed in terms of x². This form is crucial for further analysis of the equation and the ellipse it represents.

Analyzing the Solution: y² = 4 - (4/25)x²

The solution y² = 4 - (4/25)x² provides valuable insights into the relationship between y and x in the given equation. This form allows us to easily determine the values of y² for different values of x, which is essential for graphing the ellipse and understanding its behavior. For instance, we can see that the maximum value of y² occurs when x = 0, which gives y² = 4. This corresponds to the y-intercepts of the ellipse. Similarly, we can find the x-intercepts by setting y² = 0 and solving for x. The solution also reveals the elliptical nature of the equation, as the coefficient of x² is negative, indicating that the graph is bounded in both the x and y directions.

Common Mistakes to Avoid

When isolating variables in equations, it's crucial to avoid common mistakes that can lead to incorrect solutions. One frequent error is failing to apply operations to both sides of the equation. For example, when subtracting 4x² from the left side, it's essential to also subtract it from the right side to maintain equality. Another common mistake is incorrectly simplifying fractions. When dividing (100 - 4x²) by 25, it's crucial to divide both 100 and 4x² by 25. Overlooking this can lead to an incorrect expression for y². Additionally, be mindful of the order of operations. Ensure that you perform operations in the correct sequence, following the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). By being aware of these common pitfalls, you can enhance your accuracy and confidence in algebraic manipulations.

Alternative Approaches and Techniques

While the step-by-step method outlined above is a standard approach, there are alternative techniques that can be used to isolate y² in the equation 4x² + 25y² = 100. One such technique involves rearranging the equation in a slightly different order. Instead of immediately subtracting 4x², we could first divide the entire equation by 25. This would give us (4/25)x² + y² = 4. Then, we can subtract (4/25)x² from both sides to isolate y². This approach yields the same solution but demonstrates the flexibility of algebraic manipulations. Another technique involves using the concept of inverse operations. Each operation has an inverse that undoes it. For example, the inverse of addition is subtraction, and the inverse of multiplication is division. By applying inverse operations strategically, we can systematically isolate the desired variable. Exploring these alternative approaches can deepen your understanding of algebraic principles and enhance your problem-solving skills.

Real-World Applications of Isolating Variables

The skill of isolating variables extends far beyond textbook problems and finds numerous applications in real-world scenarios. In physics, for example, equations are used to describe the motion of objects, the behavior of electric circuits, and the properties of light. Isolating variables in these equations allows physicists to predict the outcome of experiments, design new technologies, and understand the fundamental laws of nature. In engineering, equations are used to design bridges, buildings, and machines. Isolating variables allows engineers to calculate the stresses and strains on structures, optimize designs for efficiency, and ensure the safety of their creations. In economics, equations are used to model supply and demand, predict market trends, and analyze economic policies. Isolating variables allows economists to understand the relationships between different economic factors and make informed decisions. These are just a few examples of how the ability to isolate variables is essential in various fields.

Practice Problems and Further Exploration

To solidify your understanding of isolating variables, it's essential to practice with a variety of problems. Try isolating different variables in different equations, and explore how the process changes depending on the equation's structure. For instance, you could try isolating x² in the equation 4x² + 25y² = 100 or isolating variables in linear equations, quadratic equations, and systems of equations. Additionally, delve deeper into the mathematical concepts related to ellipses and conic sections. Understanding the properties of these shapes will provide a richer context for the equations you're manipulating. Explore online resources, textbooks, and mathematical software to expand your knowledge and skills. The more you practice and explore, the more confident and proficient you will become in algebraic manipulations.

Conclusion: Mastering the Art of Variable Isolation

Isolating y² in the equation 4x² + 25y² = 100 is a fundamental exercise in algebraic manipulation. Through a step-by-step approach, we've demonstrated how to isolate y² and express it in terms of x². This process not only provides the solution to the problem but also reinforces the broader concept of manipulating equations to gain insights into the relationships between variables. The ability to isolate variables is a crucial skill in mathematics and its applications, enabling us to solve problems, analyze relationships, and make informed decisions. By understanding the principles and techniques involved, and by practicing diligently, you can master the art of variable isolation and unlock a deeper understanding of the mathematical world.

The correct answer is D. y² = 4 - (4/25)x²