-5/y+7/4y=13/y-40

Solution for -5/y+7/4y=13/y-40 equation:

-5/y+7/4y=13/y-40

We move all terms to the left:

-5/y+7/4y-(13/y-40)=0


Domain of the equation: y!=0

y∈R



Domain of the equation: 4y!=0

y!=0/4

y!=0

y∈R



Domain of the equation: y-40)!=0

y∈R


We get rid of parentheses

-5/y+7/4y-13/y+40=0

We calculate fractions


(-52y-5)/4y^2+7y/4y^2+40=0

We multiply all the terms by the denominator


(-52y-5)+7y+40*4y^2=0

We add all the numbers together, and all the variables

7y+(-52y-5)+40*4y^2=0

Wy multiply elements

160y^2+7y+(-52y-5)=0

We get rid of parentheses

160y^2+7y-52y-5=0

We add all the numbers together, and all the variables

160y^2-45y-5=0

a = 160; b = -45; c = -5;
Δ = b2-4ac
Δ = -452-4·160·(-5)
Δ = 5225
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{5225}=\sqrt{25*209}=\sqrt{25}*\sqrt{209}=5\sqrt{209}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-45)-5\sqrt{209}}{2*160}=\frac{45-5\sqrt{209}}{320}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-45)+5\sqrt{209}}{2*160}=\frac{45+5\sqrt{209}}{320}
$