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

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

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

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R



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

x∈R


We get rid of parentheses

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

We calculate fractions


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

We add all the numbers together, and all the variables

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

We add all the numbers together, and all the variables

3x/3x^2+(-1x)/3x^2-4=0

We multiply all the terms by the denominator


3x+(-1x)-4*3x^2=0

Wy multiply elements

-12x^2+3x+(-1x)=0

We get rid of parentheses

-12x^2+3x-1x=0

We add all the numbers together, and all the variables

-12x^2+2x=0

a = -12; b = 2; c = 0;
Δ = b2-4ac
Δ = 22-4·(-12)·0
Δ = 4
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

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