1/3y-3=5/6y+10

Solution for 1/3y-3=5/6y+10 equation:

1/3y-3=5/6y+10

We move all terms to the left:

1/3y-3-(5/6y+10)=0


Domain of the equation: 3y!=0

y!=0/3

y!=0

y∈R



Domain of the equation: 6y+10)!=0

y∈R


We get rid of parentheses

1/3y-5/6y-10-3=0

We calculate fractions


6y/18y^2+(-15y)/18y^2-10-3=0

We add all the numbers together, and all the variables

6y/18y^2+(-15y)/18y^2-13=0

We multiply all the terms by the denominator


6y+(-15y)-13*18y^2=0

Wy multiply elements

-234y^2+6y+(-15y)=0

We get rid of parentheses

-234y^2+6y-15y=0

We add all the numbers together, and all the variables

-234y^2-9y=0

a = -234; b = -9; c = 0;
Δ = b2-4ac
Δ = -92-4·(-234)·0
Δ = 81
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{81}=9$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-9}{2*-234}=\frac{0}{-468}
=0
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+9}{2*-234}=\frac{18}{-468}
=-1/26
$