3y-5=1/6y+4

Solution for 3y-5=1/6y+4 equation:

3y-5=1/6y+4

We move all terms to the left:

3y-5-(1/6y+4)=0


Domain of the equation: 6y+4)!=0

y∈R


We get rid of parentheses

3y-1/6y-4-5=0

We multiply all the terms by the denominator


3y*6y-4*6y-5*6y-1=0

Wy multiply elements

18y^2-24y-30y-1=0

We add all the numbers together, and all the variables

18y^2-54y-1=0

a = 18; b = -54; c = -1;
Δ = b2-4ac
Δ = -542-4·18·(-1)
Δ = 2988
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:

$\sqrt{\Delta}=\sqrt{2988}=\sqrt{36*83}=\sqrt{36}*\sqrt{83}=6\sqrt{83}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-54)-6\sqrt{83}}{2*18}=\frac{54-6\sqrt{83}}{36}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-54)+6\sqrt{83}}{2*18}=\frac{54+6\sqrt{83}}{36}
$