Question

Eliminate $x,y$ from the equations $x^{2}y=a,x(x+y)=b,2x+y=c$.

Answer 1

$2x+y=c$

$\Rightarrow y=c-2x$

Also, $x(x+y)=b$

$\Rightarrow x^2+x(c-2x)=b$

$\Rightarrow c-x^2=b$

$\Rightarrow x^2=c-b$

$\Rightarrow x=\sqrt{c-b}$

Hence, $y=c-2\sqrt{c-b}$

Now, $x^{2}y=a$

$\Rightarrow (c-b)(2-\sqrt{c-b})=a$

$\Rightarrow \sqrt{c-b}=2-\frac{a}{c-b}$

$\Rightarrow c-b=(2-\frac{a}{c-b}{)}^2$

$\Rightarrow (c-b{)}^3=(2c-b-a{)}^2$