1/4y-3=5-3y

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

1/4y-3=5-3y

We move all terms to the left:

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


Domain of the equation: 4y!=0

y!=0/4

y!=0

y∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

12y^2-20y-12y+1=0

We add all the numbers together, and all the variables

12y^2-32y+1=0

a = 12; b = -32; c = +1;
Δ = b2-4ac
Δ = -322-4·12·1
Δ = 976
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:

$\sqrt{\Delta}=\sqrt{976}=\sqrt{16*61}=\sqrt{16}*\sqrt{61}=4\sqrt{61}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-32)-4\sqrt{61}}{2*12}=\frac{32-4\sqrt{61}}{24}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-32)+4\sqrt{61}}{2*12}=\frac{32+4\sqrt{61}}{24}
$