2/3y-1/(-3+3y)=2

Solution for 2/3y-1/(-3+3y)=2 equation:

2/3y-1/(-3+3y)=2

We move all terms to the left:

2/3y-1/(-3+3y)-(2)=0


Domain of the equation: 3y!=0

y!=0/3

y!=0

y∈R



Domain of the equation: (-3+3y)!=0

We move all terms containing y to the left, all other terms to the right

3y!=3

y!=3/3

y!=1

y∈R


We add all the numbers together, and all the variables

2/3y-1/(3y-3)-2=0

We calculate fractions


(6y-6)/(9y^2-9y)+(-3y)/(9y^2-9y)-2=0

We multiply all the terms by the denominator


(6y-6)+(-3y)-2*(9y^2-9y)=0

We multiply parentheses

-18y^2+(6y-6)+(-3y)+18y=0

We get rid of parentheses

-18y^2+6y-3y+18y-6=0

We add all the numbers together, and all the variables

-18y^2+21y-6=0

a = -18; b = 21; c = -6;
Δ = b2-4ac
Δ = 212-4·(-18)·(-6)
Δ = 9
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{9}=3$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(21)-3}{2*-18}=\frac{-24}{-36}
=2/3
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(21)+3}{2*-18}=\frac{-18}{-36}
=1/2
$