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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



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

x∈R


We get rid of parentheses

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

We calculate fractions


3x/6x^2+(-4x)/6x^2-1-5=0

We add all the numbers together, and all the variables

3x/6x^2+(-4x)/6x^2-6=0

We multiply all the terms by the denominator


3x+(-4x)-6*6x^2=0

Wy multiply elements

-36x^2+3x+(-4x)=0

We get rid of parentheses

-36x^2+3x-4x=0

We add all the numbers together, and all the variables

-36x^2-1x=0

a = -36; b = -1; c = 0;
Δ = b2-4ac
Δ = -12-4·(-36)·0
Δ = 1
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1}=1$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-1}{2*-36}=\frac{0}{-72}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+1}{2*-36}=\frac{2}{-72}
=-1/36
$