1/2y=3y-5

Solution for 1/2y=3y-5 equation:

1/2y=3y-5

We move all terms to the left:

1/2y-(3y-5)=0


Domain of the equation: 2y!=0

y!=0/2

y!=0

y∈R


We get rid of parentheses

1/2y-3y+5=0

We multiply all the terms by the denominator


-3y*2y+5*2y+1=0

Wy multiply elements

-6y^2+10y+1=0

a = -6; b = 10; c = +1;
Δ = b2-4ac
Δ = 102-4·(-6)·1
Δ = 124
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:

$\sqrt{\Delta}=\sqrt{124}=\sqrt{4*31}=\sqrt{4}*\sqrt{31}=2\sqrt{31}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{31}}{2*-6}=\frac{-10-2\sqrt{31}}{-12}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{31}}{2*-6}=\frac{-10+2\sqrt{31}}{-12}
$