x-5=1/3x-1

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

x-5=1/3x-1

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

3x^2+3x-15x-1=0

We add all the numbers together, and all the variables

3x^2-12x-1=0

a = 3; b = -12; c = -1;
Δ = b2-4ac
Δ = -122-4·3·(-1)
Δ = 156
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:

$\sqrt{\Delta}=\sqrt{156}=\sqrt{4*39}=\sqrt{4}*\sqrt{39}=2\sqrt{39}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{39}}{2*3}=\frac{12-2\sqrt{39}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{39}}{2*3}=\frac{12+2\sqrt{39}}{6}
$