7/7x+1/3x=1+5/3x

Solution for 7/7x+1/3x=1+5/3x equation:

7/7x+1/3x=1+5/3x

We move all terms to the left:

7/7x+1/3x-(1+5/3x)=0


Domain of the equation: 7x!=0

x!=0/7

x!=0

x∈R



Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 3x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

7/7x+1/3x-(5/3x+1)=0

We get rid of parentheses

7/7x+1/3x-5/3x-1=0

We calculate fractions


21x/21x^2+(-35x+1)/21x^2-1=0

We multiply all the terms by the denominator


21x+(-35x+1)-1*21x^2=0

Wy multiply elements

-21x^2+21x+(-35x+1)=0

We get rid of parentheses

-21x^2+21x-35x+1=0

We add all the numbers together, and all the variables

-21x^2-14x+1=0

a = -21; b = -14; c = +1;
Δ = b2-4ac
Δ = -142-4·(-21)·1
Δ = 280
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:

$\sqrt{\Delta}=\sqrt{280}=\sqrt{4*70}=\sqrt{4}*\sqrt{70}=2\sqrt{70}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{70}}{2*-21}=\frac{14-2\sqrt{70}}{-42}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{70}}{2*-21}=\frac{14+2\sqrt{70}}{-42}
$