.5y=1/3y+2

Solution for .5y=1/3y+2 equation:

.5y=1/3y+2

We move all terms to the left:

.5y-(1/3y+2)=0


Domain of the equation: 3y+2)!=0

y∈R


We get rid of parentheses

.5y-1/3y-2=0

We multiply all the terms by the denominator


(.5y)*3y-2*3y-1=0

We add all the numbers together, and all the variables

(+.5y)*3y-2*3y-1=0

We multiply parentheses

3y^2-2*3y-1=0

Wy multiply elements

3y^2-6y-1=0

a = 3; b = -6; c = -1;
Δ = b2-4ac
Δ = -62-4·3·(-1)
Δ = 48
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:

$\sqrt{\Delta}=\sqrt{48}=\sqrt{16*3}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-4\sqrt{3}}{2*3}=\frac{6-4\sqrt{3}}{6}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+4\sqrt{3}}{2*3}=\frac{6+4\sqrt{3}}{6}
$