Can this systems of equations be solved? There is a system of equations as follow:
$$\left[\begin{matrix}x\\ y\\ \end{matrix} \right] = \left[\begin{matrix}a& b\\b & c\\\end{matrix} \right]\left[\begin{matrix}\frac{1}{x}\\ \frac{1}{y}\\ \end{matrix} \right]$$
Given $ \left[\begin{matrix}a& b\\b & c\\\end{matrix} \right]$, can this system of equations be solved for $x$ and $y$?

And how to solve the extended case?
$$\left[\begin{matrix}x_1\\ x_2\\ \vdots \\x_n \end{matrix} \right] =
 \left[\begin{matrix}a_{11}& a_{12} & \dots &a_{1n}\\a_{12} & a_{22} & \dots  &a_{2n}\\ \vdots & \vdots & &\vdots \\a_{1n} & a_{2n} & \dots &a_{nn}\end{matrix} \right]\left[\begin{matrix}\frac{1}{x_1}\\ \frac{1}{x_2}\\ \vdots \\ \frac{1}{x_n}\\ \end{matrix} \right]$$

After pie314271's answer
The equation $$b^2(x^2-a)+c(x^2-a)^2=b^2x^2$$ reduces to $$a\left(a c- b^2\right)-2 a c x^2+c x^4=0$$ which is just a quadratic in $x^2$ and the solutions are given by $$x=\pm\sqrt{a\pm\frac{\sqrt{a}}{\sqrt{c}}\,b}$$ So, the whole set of solutions is
$$\left\{x= -\sqrt{a-\frac{\sqrt{a} b}{\sqrt{c}}},y= \frac{\sqrt{c}
   \sqrt{a-\frac{\sqrt{a} b}{\sqrt{c}}}}{\sqrt{a}}\right\},\left\{x=
   \sqrt{a-\frac{\sqrt{a} b}{\sqrt{c}}},y= -\frac{\sqrt{c} \sqrt{a-\frac{\sqrt{a}
   b}{\sqrt{c}}}}{\sqrt{a}}\right\},\left\{x=-\sqrt{a+\frac{\sqrt{a}
   b}{\sqrt{c}}},y= -\frac{\sqrt{c} \sqrt{a+\frac{\sqrt{a}
   b}{\sqrt{c}}}}{\sqrt{a}}\right\},\left\{x= \sqrt{a+\frac{\sqrt{a}
   b}{\sqrt{c}}},y=\frac{\sqrt{c} \sqrt{a+\frac{\sqrt{a}
   b}{\sqrt{c}}}}{\sqrt{a}}\right\}$$
