%2F(x%5E2-xy%2By%5E2)%3D3%2F7.jpeg "(x+y)/(x^2-xy+y^2)=3/7")
Solving the Equation: (x+y)/(x^2-xy+y^2) = 3/7
This problem involves simplifying a rational expression and solving for the relationship between x and y. Let's break it down step-by-step.
1. Simplifying the Expression
The denominator, x² - xy + y², resembles a special form. However, we can't directly factor it like a difference of squares. Let's focus on the numerator:
- (x + y): This expression cannot be further simplified.
2. Cross-Multiplication
To solve for x and y, we'll cross-multiply the equation:
- **7(x + y) = 3(x² - xy + y²) **
3. Expanding and Rearranging
Expanding both sides of the equation:
- **7x + 7y = 3x² - 3xy + 3y² **
Rearranging to form a quadratic equation:
- **3x² - 3xy + 3y² - 7x - 7y = 0 **
4. Solutions and Interpretation
The equation above represents a quadratic equation in two variables. There are several ways to approach finding solutions, but it's important to recognize a few key points:
- Multiple Solutions: This equation likely has multiple solutions, meaning there are multiple pairs of x and y values that satisfy the original equation.
- Relationship: The equation defines a relationship between x and y. It's a more complex relationship than a simple linear equation.
5. Further Analysis
To find specific solutions, you would need to use techniques like:
- Factoring: If possible, factor the quadratic equation.
- Quadratic Formula: Applying the quadratic formula to solve for either x or y (treating the other variable as a constant).
- Graphical Methods: Plotting the equation as a curve and identifying intersection points with the axes.
Conclusion
While solving for x and y explicitly involves more advanced techniques, understanding the steps to simplify and rearrange the equation provides a foundation for further analysis and finding solutions. Remember, the equation represents a relationship between x and y, and there may be multiple pairs of values that satisfy the original condition.