1/x=2/3x+1

Solution for 1/x=2/3x+1 equation:

1/x=2/3x+1

We move all terms to the left:

1/x-(2/3x+1)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 3x+1)!=0

x∈R


We get rid of parentheses

1/x-2/3x-1=0

We calculate fractions


3x/3x^2+(-2x)/3x^2-1=0

We multiply all the terms by the denominator


3x+(-2x)-1*3x^2=0

Wy multiply elements

-3x^2+3x+(-2x)=0

We get rid of parentheses

-3x^2+3x-2x=0

We add all the numbers together, and all the variables

-3x^2+x=0

a = -3; b = 1; c = 0;
Δ = b2-4ac
Δ = 12-4·(-3)·0
Δ = 1
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

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