1/3x-3=-x+5

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

1/3x-3=-x+5

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

3x^2-15x-9x+1=0

We add all the numbers together, and all the variables

3x^2-24x+1=0

a = 3; b = -24; c = +1;
Δ = b2-4ac
Δ = -242-4·3·1
Δ = 564
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:

$\sqrt{\Delta}=\sqrt{564}=\sqrt{4*141}=\sqrt{4}*\sqrt{141}=2\sqrt{141}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-2\sqrt{141}}{2*3}=\frac{24-2\sqrt{141}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+2\sqrt{141}}{2*3}=\frac{24+2\sqrt{141}}{6}
$