X=6+2/3y

Solution for X=6+2/3y equation:

X=6+2/3X

We move all terms to the left:

X-(6+2/3X)=0


Domain of the equation: 3X)!=0

X!=0/1

X!=0

X∈R


We add all the numbers together, and all the variables

X-(2/3X+6)=0

We get rid of parentheses

X-2/3X-6=0

We multiply all the terms by the denominator


X*3X-6*3X-2=0

Wy multiply elements

3X^2-18X-2=0

a = 3; b = -18; c = -2;
Δ = b2-4ac
Δ = -182-4·3·(-2)
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{87}}{2*3}=\frac{18-2\sqrt{87}}{6}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{87}}{2*3}=\frac{18+2\sqrt{87}}{6}
$