Problem solving a system of equations

Question

How can I solve this equation system for x and y?

$\frac{xy}{x+1} = a$

$\frac{xy^2}{x+2} = b$

$a>0$ $b>0$

I've tried to isolate $y$ of both equations and then equal the result to solve $x$ but then I can't isolate $x$. I've also tried to combine the two equations but I get nothing.

Answer 1

Note that $$ y=\frac{a(x+1)}{x}\ \text{and}\ y^2=\frac{b(x+2)}{x} $$ Therefore, $$ \frac{a^2(x+1)^2}{x^2}=\frac{b(x+2)}{x} $$ that is $$a^2(x+1)^2=bx(x+2)$$ Hence $$ (a^2-b)x^2+2(a^2-b)x+a^2=0 $$ Now solve this quadratic equation.

Answer 2

Square the first equation \begin{eqnarray*} \frac{x^2y^2}{(x+1)^2}=a^2. \end{eqnarray*} Now rewrite the second equation \begin{eqnarray*} \frac{x^2y^2}{(x+1)^2} \times\frac{(x+1)^2}{x(x+2)} =b. \end{eqnarray*} So we have the quadratic \begin{eqnarray*} a^2(x+1)^2=bx(x+2) \end{eqnarray*} which is easily solved.