1/3y+7=3y-1

Solution for 1/3y+7=3y-1 equation:

1/3y+7=3y-1

We move all terms to the left:

1/3y+7-(3y-1)=0


Domain of the equation: 3y!=0

y!=0/3

y!=0

y∈R


We get rid of parentheses

1/3y-3y+1+7=0

We multiply all the terms by the denominator


-3y*3y+1*3y+7*3y+1=0

Wy multiply elements

-9y^2+3y+21y+1=0

We add all the numbers together, and all the variables

-9y^2+24y+1=0

a = -9; b = 24; c = +1;
Δ = b2-4ac
Δ = 242-4·(-9)·1
Δ = 612
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:

$\sqrt{\Delta}=\sqrt{612}=\sqrt{36*17}=\sqrt{36}*\sqrt{17}=6\sqrt{17}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-6\sqrt{17}}{2*-9}=\frac{-24-6\sqrt{17}}{-18}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+6\sqrt{17}}{2*-9}=\frac{-24+6\sqrt{17}}{-18}
$