6+1/3y=10+12y

Solution for 6+1/3y=10+12y equation:

6+1/3y=10+12y

We move all terms to the left:

6+1/3y-(10+12y)=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-(12y+10)+6=0

We get rid of parentheses

1/3y-12y-10+6=0

We multiply all the terms by the denominator


-12y*3y-10*3y+6*3y+1=0

Wy multiply elements

-36y^2-30y+18y+1=0

We add all the numbers together, and all the variables

-36y^2-12y+1=0

a = -36; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·(-36)·1
Δ = 288
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:

$\sqrt{\Delta}=\sqrt{288}=\sqrt{144*2}=\sqrt{144}*\sqrt{2}=12\sqrt{2}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-12\sqrt{2}}{2*-36}=\frac{12-12\sqrt{2}}{-72}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+12\sqrt{2}}{2*-36}=\frac{12+12\sqrt{2}}{-72}
$