Quick tangent question, please HELP? I am desperate!!?

Question

Determine the equation of the tangent to the curve 2/x + 2/y = 5 at point (1, 2/3).

Answer 1

2/y + 2/x = 5
2/y = 5 - 2/x
2/y = (5x - 2)/x
y/2 = x/(5x-2)
y = 2x/(5x-2)
y = 2x(5x-2)^-1
dy/dx = 2(5x-2)^-1 + 2x(-1)(5x-2)^-2.(5)
dy/dx = 2(5x-2)^-1 - 10x(5x-2)^-2
dy/dx|x=1 = 2(3)^-1 - 10(1)(3)^-2
dy/dx|x=1 = 2/3 - 10/9
dy/dx|x=1 = -4/9

y = mx + c
2/3 = (-4/9)(1) + c
c = 2/3 + 4/9
c = 10/9

Equation of tangent:
y = (-4/9)x + 10/9
9y = -4x + 10
4x + 9y = 10

Answer 2

`By implicit differentiation
-2/x^2-2/y^2*y´=0 so y´= -y^2/x^2= -4/9 the equation of the tangent is
y-2/3=-4/9(x-1)
y=-4/9x+5/3

Answer 3

Differentiate the equation with respect to x.

-2/x^2 - 2y'/y^2 = 0

y' = - [y/x]^2 = -4/9 = m

y = -4x/9 +b

2/3 = -4/9 + b --> b = 10/9

y = -4x/9 + 10/9 or 4x + 9y = 10

Answer 4

-2/x² - 2/y² . dy/dx = 0

-2 - 9/2 dy/dx = 0

dy/dx = -4/9

(y - 2/3)/ (x - 1) = -4/9

9y + 4x - 10 = 0