Math... x = y-1/y-2 Solve for Y?

Question

Please breakdown the problem and show each step.

Answer 1

x(y-2) = y - 1
xy - 2x = y - 1
1 - 2x = y - xy
1 - 2x = y(1 - x)
y = (1 - 2x)/(1-x)

Answer 2

Multiply through by y: xy = y^2 - 1 - 2y, or 0 = y^2 - (2 + x) y -1. Use the quadratic equation: y = [2 + x +/- sqrt((2 + x)^2 + 4))]/2 = [2 + x +/- sqrt(x^2 + 4x + 8)]/2.

Do you know anything more about the relation between x and y?

If your problem is
x = y - 1/(y - 2), then x(y - 2) = y(y - 2) - 1, or y^2 + y(-2 - x) - 1 + 2x. Now the Q.F yields y = [2 + x +/- sqrt(x^2 - 4x + 8)]/2. Not much different.