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Question 1209749: If (x,y) satisfies the simultaneous equations
3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20 = 0.
x^2 - y^2 = 7 + 4xy
where x and y may be complex numbers, determine all possible values of y^2.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this system of equations and find the possible values of y²:
**1. Rewrite the first equation:**
Notice that several terms in the first equation have a common factor. We can rearrange and factor:
* 3xy - 4x² + 18y - 24x + 5x²y - 8y³ + 20 = 0
* (5x² + 3x - 8y²)y - (4x² + 24x - 18y - 20) = 0
* y(5x² + 3x - 8y²) - (4x² + 24x - 18y - 20) = 0
**2. Rewrite the second equation:**
* x² - y² = 7 + 4xy
* x² - 4xy - y² = 7
**3. Combine the equations:**
This is where the problem gets a bit tricky. There isn't a straightforward substitution. Let's try to manipulate the equations to eliminate x.
From the second equation, isolate x²:
x² = y² + 7 + 4xy
Substitute this expression for x² into the first equation:
y(5(y² + 7 + 4xy) + 3x - 8y²) - (4(y² + 7 + 4xy) + 24x - 18y - 20) = 0
y(5y² + 35 + 20xy + 3x - 8y²) - (4y² + 28 + 16xy + 24x - 18y - 20) = 0
y(-3y² + 20xy + 35 + 3x) - (4y² + 16xy + 8 + 24x - 18y) = 0
-3y³ + 20xy² + 35y + 3xy - 4y² - 16xy - 8 - 24x + 18y = 0
-3y³ + 16xy² - 4y² + 3xy - 16xy + 53y - 24x - 8 = 0
-3y³ + 16xy² - 4y² - 13xy + 53y - 24x - 8 = 0
Let's try a different approach. We can rewrite the first equation as:
5x²y - 8y³ + 3xy - 4x² + 18y - 24x + 20 = 0
And the second equation as:
x² - 4xy - y² = 7
Multiply the second equation by 5y:
5x²y - 20xy² - 5y³ = 35y
Substitute this into the first equation:
5x²y - 8y³ + 3xy - 4x² + 18y - 24x + 20 = 0
(35y + 20xy²) - 8y³ + 3xy - 4x² + 18y - 24x + 20 = 0
-8y³ + 20xy² + 3xy - 4x² + 53y - 24x + 20 = 0
This is still quite complex. Let's consider the second equation again: x² - 4xy - y² = 7. We can complete the square for x:
(x - 2y)² - 4y² - y² = 7
(x - 2y)² - 5y² = 7
(x - 2y)² = 7 + 5y²
This doesn't seem to lead to a simple solution.
Given the complexity, it's possible there's a more elegant approach or a trick I'm missing. However, without further simplification, solving for y² directly is quite challenging. A computational approach using a symbolic solver might be the most practical way to proceed. Such a solver would likely give numerical solutions for y², which might be complex.
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