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

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

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

We move all terms to the left:

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


Domain of the equation: (y-1)!=0

We move all terms containing y to the left, all other terms to the right

y!=1

y∈R



Domain of the equation: y)!=0

y!=0/1

y!=0

y∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We calculate fractions


5y/(y^2-1y)+(-4y+4)/(y^2-1y)-3=0

We multiply all the terms by the denominator


5y+(-4y+4)-3*(y^2-1y)=0

We multiply parentheses

-3y^2+5y+(-4y+4)+3y=0

We get rid of parentheses

-3y^2+5y-4y+3y+4=0

We add all the numbers together, and all the variables

-3y^2+4y+4=0

a = -3; b = 4; c = +4;
Δ = b2-4ac
Δ = 42-4·(-3)·4
Δ = 64
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{64}=8$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-8}{2*-3}=\frac{-12}{-6}
=+2
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+8}{2*-3}=\frac{4}{-6}
=-2/3
$