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

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

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

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



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

x∈R


We get rid of parentheses

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

We calculate fractions


3x/3x^2+(-15x)/3x^2+1-2=0

We add all the numbers together, and all the variables

3x/3x^2+(-15x)/3x^2-1=0

We multiply all the terms by the denominator


3x+(-15x)-1*3x^2=0

Wy multiply elements

-3x^2+3x+(-15x)=0

We get rid of parentheses

-3x^2+3x-15x=0

We add all the numbers together, and all the variables

-3x^2-12x=0

a = -3; b = -12; c = 0;
Δ = b2-4ac
Δ = -122-4·(-3)·0
Δ = 144
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{144}=12$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-12}{2*-3}=\frac{0}{-6}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+12}{2*-3}=\frac{24}{-6}
=-4
$