1/3n=3/n-1

Solution for 1/3n=3/n-1 equation:

1/3n=3/n-1

We move all terms to the left:

1/3n-(3/n-1)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R



Domain of the equation: n-1)!=0

n∈R


We get rid of parentheses

1/3n-3/n+1=0

We calculate fractions


n/3n^2+(-9n)/3n^2+1=0

We multiply all the terms by the denominator


n+(-9n)+1*3n^2=0

Wy multiply elements

3n^2+n+(-9n)=0

We get rid of parentheses

3n^2+n-9n=0

We add all the numbers together, and all the variables

3n^2-8n=0

a = 3; b = -8; c = 0;
Δ = b2-4ac
Δ = -82-4·3·0
Δ = 64
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{64}=8$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-8}{2*3}=\frac{0}{6}
=0
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+8}{2*3}=\frac{16}{6}
=2+2/3
$