I need some help with Solving Equations with Rational Expressions?

1)
x/3 - 2/3 = 1/x
(x - 2)/3 = 1/x
x - 2 = 3(1/x)
x - 2 = 3/x
x(x - 2) = 3
x^2 - 2x = 3
x^2 - 2x - 3 = 0
x^2 + x - 3x - 3 = 0
(x^2 + x) - (3x + 3) = 0
x(x + 1) - 3(x + 1) = 0
(x + 1)(x - 3) = 0

x + 1 = 0
x = -1

x - 3 = 0
x = 3

∴ x = -1 , 3

= = = = = = = =

2)
(y + 2)/y = 1/(y - 5)
y + 2 = y[1/(y - 5)]
y + 2 = y/(y - 5)
(y - 5)(y + 2) = y
y^2 - 5y + 2y - 10 - y = 0
y^2 - 3y - y - 10 = 0
y^2 - 4y - 10 = 0
y = [-b ±√(b^2 - 4ac)]/2a

a = 1
b = -4
c = -10

y = [4 ±√(16 + 40)]/2
y = [4 ±√56]/2
y = [4 ±√(2^2 * 2 * 7)]/2
y = [4 ±2√(2 * 7)]/2
y = [4 ±2√14]/2
y = 2 ±√14

∴ y = 2 ±√14

1st problem:
x/3 - 2/3 = 1/x
x/3 - 1/x = 2/3
(x² - 3)/3x = 2/3
x² - 3 = 2x
x² - 2x = 3
x² - x = 3 + 1
(x - 1)² = 4
x - 1 = 2

x = 2 + 1, x = 3
x = - 2 + 1, x = - 1

Answer: x = 3, - 1

Proof (x = 3):
3/3 - 2/3 = 1/3
1/3 = 1/3

Proof (x = - 1):
- 1/3 - 2/3 = 1/- 1
- 3/3 = - 1
- 1 = - 1

2nd problem:
(y + 2)/y = 1/(y - 5)
(y + 2)(y - 5) = y
y² - 5y + 2y - 10 = y
y² - 4y = 10
y² - 2y = 10 + 2²
y² - 2y = 10 + 4
(y - 2)² = 14
y - 2 = 3.7416573

y = 3.7416573 + 2, y = 5.7416573
y = - 3.7416573 + 2, y = - 1.7416573

Answer: y = 5.7416573, y = - 1.7416573

Proof (y = 5.7416573):
(5.7416573 + 2)/5.7416573 = 1/(5.7416573 - 5)
7.7416573/5.7416573 = 1/0.7416573
1.34833 = 1.34833

Proof (y = - 1.7416573):
(- 1.7416573 + 2)/- 1.7416573 = 1/(- 1.7416573 - 5)
0.2583427/- 1.7416573 = 1/- 6.7416573
- 0.14833 = 0.14833

Question 1
x/3 - 2/3 = 1/x
x² - 2x = 3
x² - 2x - 3 = 0
(x - 3) (x + 1) = 0
x = 3 , x = - 1

Question 2
As given, the question is INCORRECT.
It SHOULD be shown as :-
(y + 2) / y = 1 / (y - 5)
(y + 2)(y - 5) = y
y² - 3y - 10 = y
y² - 4y - 10 = 0
y = [ 4 ± √ (16 + 40 ) ] / 2
y = [ 4 ± √ (16 + 40 ) ] / 2
y = [ 4 ± √ (56) ] / 2
y = [ 4 ± 2√ (14) ] / 2
y = 2 ± √14

1. (x-2)/3 = 1/x
x(x-2) = 3
x^2-2x-3=0
(x-3)(x+1)=0
x = 3 & x = -1

2. (y^2+2)/y=(1-5y)/y
y^2+5y+1=0
and then solve the equation.