1/6x=2/3x+1

Solution for 1/6x=2/3x+1 equation:

1/6x=2/3x+1

We move all terms to the left:

1/6x-(2/3x+1)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R



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

x∈R


We get rid of parentheses

1/6x-2/3x-1=0

We calculate fractions


3x/18x^2+(-12x)/18x^2-1=0

We multiply all the terms by the denominator


3x+(-12x)-1*18x^2=0

Wy multiply elements

-18x^2+3x+(-12x)=0

We get rid of parentheses

-18x^2+3x-12x=0

We add all the numbers together, and all the variables

-18x^2-9x=0

a = -18; b = -9; c = 0;
Δ = b2-4ac
Δ = -92-4·(-18)·0
Δ = 81
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{81}=9$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-9}{2*-18}=\frac{0}{-36}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+9}{2*-18}=\frac{18}{-36}
=-1/2
$