3y+31=1/2y-16

Solution for 3y+31=1/2y-16 equation:

3y+31=1/2y-16

We move all terms to the left:

3y+31-(1/2y-16)=0


Domain of the equation: 2y-16)!=0

y∈R


We get rid of parentheses

3y-1/2y+16+31=0

We multiply all the terms by the denominator


3y*2y+16*2y+31*2y-1=0

Wy multiply elements

6y^2+32y+62y-1=0

We add all the numbers together, and all the variables

6y^2+94y-1=0

a = 6; b = 94; c = -1;
Δ = b2-4ac
Δ = 942-4·6·(-1)
Δ = 8860
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:

$\sqrt{\Delta}=\sqrt{8860}=\sqrt{4*2215}=\sqrt{4}*\sqrt{2215}=2\sqrt{2215}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(94)-2\sqrt{2215}}{2*6}=\frac{-94-2\sqrt{2215}}{12}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(94)+2\sqrt{2215}}{2*6}=\frac{-94+2\sqrt{2215}}{12}
$