Solve the system of equations for $x$ and $y$:
$$ \left(\frac{x}{8-2y}\right)^2 - \left(\frac{y}{-4}\right)^2=4 \\ \frac{x}{8-2y} + \frac{x}{-2}=1 $$
I used Lagrange multipliers with multiple constraints to get to this point. I was told to use maple but it is giving out a very strange answer. Apparently the computations are feasible by hand but I am not sure how to go about solving this for all potential values of $x$ and $y$. Help!
Hint:
If you solve the second equation for $y$, you get:
$$y = \dfrac{3 x+8}{x+2}$$
Substituting this into the first equation, you get an expression in $x$, as:
$$4 x^4+32 x^3+23 x^2-176 x - 256 = 0$$
You can either find the exact roots to this quartic equation or use Netwon's Method or equivalent for numerical results.
This gives you two real and two imaginary roots, but I will assume you only care about the real ones.
All that is left after finding those two real (unless you also want imaginary) $x$ roots is to substitute them into your equation for the $y's$, resulting in:
$$(x, y) = (-6.1942314281353956211,~2.5231545911883245705)$$
$$(x, y) = (2.3578739564003400098, 3.4589393865012162370)$$