1/9x-8/9x=1/3x-1

Solution for 1/9x-8/9x=1/3x-1 equation:

1/9x-8/9x=1/3x-1

We move all terms to the left:

1/9x-8/9x-(1/3x-1)=0


Domain of the equation: 9x!=0

x!=0/9

x!=0

x∈R



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

x∈R


We get rid of parentheses

1/9x-8/9x-1/3x+1=0

We calculate fractions


(-24x+1)/27x^2+(-9x)/27x^2+1=0

We multiply all the terms by the denominator


(-24x+1)+(-9x)+1*27x^2=0

Wy multiply elements

27x^2+(-24x+1)+(-9x)=0

We get rid of parentheses

27x^2-24x-9x+1=0

We add all the numbers together, and all the variables

27x^2-33x+1=0

a = 27; b = -33; c = +1;
Δ = b2-4ac
Δ = -332-4·27·1
Δ = 981
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{981}=\sqrt{9*109}=\sqrt{9}*\sqrt{109}=3\sqrt{109}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-33)-3\sqrt{109}}{2*27}=\frac{33-3\sqrt{109}}{54}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-33)+3\sqrt{109}}{2*27}=\frac{33+3\sqrt{109}}{54}
$