6y-2x 10x+2xy ——— = ————- 3xy 2x

Interpret as a proportion: (6y - 2x) / (3xy) = (10x + 2xy) / (2x)

Domain: x ≠ 0 and y ≠ 0 (since denominators contain 3xy and 2x).

1. Simplify each side:

• Left: (6y - 2x) / (3xy) = [2(3y - x)]/(3xy)
• Right: (10x + 2xy) / (2x) = 5 + y

So 6y - 2x = 3xy(5 + y) = 15xy + 3xy^2

2. Solve for y (treat y as the unknown, x as a parameter): 3xy^2 + 15xy - 6y + 2x = 0

This is a quadratic in y with: a = 3x, b = 15x - 6, c = 2x

Discriminant: D = b^2 - 4ac = (15x - 6)^2 - 24x^2 = 201x^2 - 180x + 36

Thus y = [-(15x - 6) ± sqrt(D)] / (2·3x) = (6 - 15x ± sqrt(201x^2 - 180x + 36)) / (6x)

3. If you prefer solving for x in terms of y: From 6y - 2x = 15xy + 3xy^2, x(3y^2 + 15y + 2) = 6y So x = 6y / (3y^2 + 15y + 2), provided 3y^2 + 15y + 2 ≠ 0.

That gives all solutions (x, y) with x ≠ 0 and y ≠ 0. If you want a specific variable solved for, tell me which one you need.