n-10=5/6n-7-13n

Solution for n-10=5/6n-7-13n equation:

n-10=5/6n-7-13n

We move all terms to the left:

n-10-(5/6n-7-13n)=0


Domain of the equation: 6n-7-13n)!=0

We move all terms containing n to the left, all other terms to the right

6n-13n)!=7

n∈R


We add all the numbers together, and all the variables

n-(-13n+5/6n-7)-10=0

We get rid of parentheses

n+13n-5/6n+7-10=0

We multiply all the terms by the denominator


n*6n+13n*6n+7*6n-10*6n-5=0

Wy multiply elements

6n^2+78n^2+42n-60n-5=0

We add all the numbers together, and all the variables

84n^2-18n-5=0

a = 84; b = -18; c = -5;
Δ = b2-4ac
Δ = -182-4·84·(-5)
Δ = 2004
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2004}=\sqrt{4*501}=\sqrt{4}*\sqrt{501}=2\sqrt{501}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{501}}{2*84}=\frac{18-2\sqrt{501}}{168}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{501}}{2*84}=\frac{18+2\sqrt{501}}{168}
$