1/6x+3=7/9x-2

Solution for 1/6x+3=7/9x-2 equation:

1/6x+3=7/9x-2

We move all terms to the left:

1/6x+3-(7/9x-2)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R



Domain of the equation: 9x-2)!=0

x∈R


We get rid of parentheses

1/6x-7/9x+2+3=0

We calculate fractions


9x/54x^2+(-42x)/54x^2+2+3=0

We add all the numbers together, and all the variables

9x/54x^2+(-42x)/54x^2+5=0

We multiply all the terms by the denominator


9x+(-42x)+5*54x^2=0

Wy multiply elements

270x^2+9x+(-42x)=0

We get rid of parentheses

270x^2+9x-42x=0

We add all the numbers together, and all the variables

270x^2-33x=0

a = 270; b = -33; c = 0;
Δ = b2-4ac
Δ = -332-4·270·0
Δ = 1089
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1089}=33$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-33)-33}{2*270}=\frac{0}{540}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-33)+33}{2*270}=\frac{66}{540}
=11/90
$