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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 6x+1)!=0

x∈R


We get rid of parentheses

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

We calculate fractions


(-72x^2)/108x^2+54x/108x^2+(-18x)/108x^2-1=0

We multiply all the terms by the denominator


(-72x^2)+54x+(-18x)-1*108x^2=0

Wy multiply elements

(-72x^2)-108x^2+54x+(-18x)=0

We get rid of parentheses

-72x^2-108x^2+54x-18x=0

We add all the numbers together, and all the variables

-180x^2+36x=0

a = -180; b = 36; c = 0;
Δ = b2-4ac
Δ = 362-4·(-180)·0
Δ = 1296
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1296}=36$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(36)-36}{2*-180}=\frac{-72}{-360}
=1/5
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(36)+36}{2*-180}=\frac{0}{-360}
=0
$