1/3y-2=2y+2-9y

Solution for 1/3y-2=2y+2-9y equation:

1/3y-2=2y+2-9y

We move all terms to the left:

1/3y-2-(2y+2-9y)=0


Domain of the equation: 3y!=0

y!=0/3

y!=0

y∈R


We add all the numbers together, and all the variables

1/3y-(-7y+2)-2=0

We get rid of parentheses

1/3y+7y-2-2=0

We multiply all the terms by the denominator


7y*3y-2*3y-2*3y+1=0

Wy multiply elements

21y^2-6y-6y+1=0

We add all the numbers together, and all the variables

21y^2-12y+1=0

a = 21; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·21·1
Δ = 60
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{15}}{2*21}=\frac{12-2\sqrt{15}}{42}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{15}}{2*21}=\frac{12+2\sqrt{15}}{42}
$