1/3x-5=2+5/6x

Solution for 1/3x-5=2+5/6x equation:

1/3x-5=2+5/6x

We move all terms to the left:

1/3x-5-(2+5/6x)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 6x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

1/3x-(5/6x+2)-5=0

We get rid of parentheses

1/3x-5/6x-2-5=0

We calculate fractions


6x/18x^2+(-15x)/18x^2-2-5=0

We add all the numbers together, and all the variables

6x/18x^2+(-15x)/18x^2-7=0

We multiply all the terms by the denominator


6x+(-15x)-7*18x^2=0

Wy multiply elements

-126x^2+6x+(-15x)=0

We get rid of parentheses

-126x^2+6x-15x=0

We add all the numbers together, and all the variables

-126x^2-9x=0

a = -126; b = -9; c = 0;
Δ = b2-4ac
Δ = -92-4·(-126)·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*-126}=\frac{0}{-252}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+9}{2*-126}=\frac{18}{-252}
=-1/14
$