1/2y+2/3y-1/6=4+1/3y

Solution for 1/2y+2/3y-1/6=4+1/3y equation:

1/2y+2/3y-1/6=4+1/3y

We move all terms to the left:

1/2y+2/3y-1/6-(4+1/3y)=0


Domain of the equation: 2y!=0

y!=0/2

y!=0

y∈R



Domain of the equation: 3y!=0

y!=0/3

y!=0

y∈R



Domain of the equation: 3y)!=0

y!=0/1

y!=0

y∈R


We add all the numbers together, and all the variables

1/2y+2/3y-(1/3y+4)-1/6=0

We get rid of parentheses

1/2y+2/3y-1/3y-4-1/6=0

We calculate fractions


(-18y^2)/216y^2+108y/216y^2+(-72y+2)/216y^2-4=0

We multiply all the terms by the denominator


(-18y^2)+108y+(-72y+2)-4*216y^2=0

Wy multiply elements

(-18y^2)-864y^2+108y+(-72y+2)=0

We get rid of parentheses

-18y^2-864y^2+108y-72y+2=0

We add all the numbers together, and all the variables

-882y^2+36y+2=0

a = -882; b = 36; c = +2;
Δ = b2-4ac
Δ = 362-4·(-882)·2
Δ = 8352
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:

$\sqrt{\Delta}=\sqrt{8352}=\sqrt{144*58}=\sqrt{144}*\sqrt{58}=12\sqrt{58}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(36)-12\sqrt{58}}{2*-882}=\frac{-36-12\sqrt{58}}{-1764}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(36)+12\sqrt{58}}{2*-882}=\frac{-36+12\sqrt{58}}{-1764}
$