1/3x-5+2/3x=7/6x+4

Solution for 1/3x-5+2/3x=7/6x+4 equation:

1/3x-5+2/3x=7/6x+4

We move all terms to the left:

1/3x-5+2/3x-(7/6x+4)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 6x+4)!=0

x∈R


We get rid of parentheses

1/3x+2/3x-7/6x-4-5=0

We calculate fractions


(12x+1)/18x^2+(-21x)/18x^2-4-5=0

We add all the numbers together, and all the variables

(12x+1)/18x^2+(-21x)/18x^2-9=0

We multiply all the terms by the denominator


(12x+1)+(-21x)-9*18x^2=0

Wy multiply elements

-162x^2+(12x+1)+(-21x)=0

We get rid of parentheses

-162x^2+12x-21x+1=0

We add all the numbers together, and all the variables

-162x^2-9x+1=0

a = -162; b = -9; c = +1;
Δ = b2-4ac
Δ = -92-4·(-162)·1
Δ = 729
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{729}=27$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-27}{2*-162}=\frac{-18}{-324}
=1/18
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+27}{2*-162}=\frac{36}{-324}
=-1/9
$