# (7x)/(x^2+2xy+y^2)+(3x)/(x^2+xy)

$\frac{7x}{{x}^{2}+2xy+y}﻿+\frac{3x}{{x}^{2}+xy}=0$
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Dominique Ferrell
$\frac{7x}{{x}^{2}+2xy+y}+\frac{3x}{{x}^{2}+xy}=0$
$\frac{7x}{{x}^{2}+2xy+y}=-\frac{3x}{{x}^{2}+xy}$
doing cross multiplication:
$7x\left({x}^{2}+xy\right)=\left({x}^{2}+2xy+y\right)\left(-3x\right)$
$7{x}^{3}+7{x}^{2}y=-3{x}^{3}-6{x}^{2}y-3xy$
isolate anything with y on one side and rest on other
$7{x}^{2}y+6{x}^{2}y+3xy=-3{x}^{3}-7{x}^{3}$
$13{x}^{2}y+3xy=-10{x}^{3}$
$y\left(13{x}^{2}+3x\right)=-10{x}^{3}$
$y=\frac{-10{x}^{3}}{13{x}^{2}+3x}$
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Bruno Thompson
$\left(7x\right)/\left({x}^{2}+2xy+y\right)+\left(3x\right)/\left({x}^{2}+xy\right)=0$
$\left(7x\right)/\left({x}^{2}+2xy+y\right)+\left(3x\right)/\left(x\left(x\right)+x\left(y\right)\right)=0$
$\left(7x\right)/\left({x}^{2}+2xy+y\right)+\left(3x\right)/\left(x\left(x+y\right)\right)=0$
Least common denominator: $x\left({x}^{2}+2xy+y\right)\left(x+y\right)$
$\left(7x\right)/\left({x}^{2}+2xy+y\right)\ast x\left({x}^{2}+2xy+y\right)\left(x+y\right)+\left(3x\right)/\left(x\left(x+y\right)\right)\ast x\left({x}^{2}+2xy+y\right)\left(x+y\right)=0\ast x\left({x}^{2}+2xy+y\right)\left(x+y\right)$
$10{x}^{3}+13{x}^{2}y+3xy=0\ast x\left({x}^{2}+2xy+y\right)\left(x+y\right)$
$10{x}^{3}+13{x}^{2}y+3xy=0$
$13{x}^{2}y+3xy=-10{x}^{3}$
$13{x}^{2}y+3xy+10{x}^{3}=0$
$x\left(10{x}^{2}\right)+x\left(13xy\right)+x\left(3y\right)=0$
$x\left(10{x}^{2}+13xy+3y\right)=0$
x=0
$10{x}^{2}+13xy+3y=0$
$13xy+3y=-10{x}^{2}$
$y\left(13x\right)+y\left(3\right)=-10{x}^{2}$
$y\left(13x+3\right)=-10{x}^{2}$
$\left(y\left(13x+3\right)\right)/\left(13x+3\right)=-\left(10{x}^{2}\right)/\left(13x+3\right)$
$y=-\left(10{x}^{2}\right)/\left(13x+3\right)$
ANSWER $y=0,-\left(10{x}^{2}\right)/\left(13x+3\right)$