I have two equations: $|c x_1 + d x_2 | = u $ and $|c x_2 - d x_1 | = v $. I know c, d, v and u. Is it possible to find out if $c x_1 + d x_2$ and $c x_2 - d x_1$ is bigger than or smaller than $0$? I have tried googling and also to solve it with wolfram alpha, but I am stuck.
Thanks in advance.
On
The linear equation $$\begin{eqnarray} c x_1 + d x_2 = U \\ - d x_1 +c x_2= V \end{eqnarray}$$ has exactly one solution if and only if its determinant $c^2+d^2\neq 0$ . For real $c$ and $d$ this is equivalent to $(c,d)\neq (0,0)$. It does not depend on $U$ and $V$. If $(c,d)\neq (0,0)$ then there is a solution for every $(U,V)\in\{(u,v),(u,-v),(-u,v),(-u,-v)\}$ .
We can't. $|c x_1 + d x_2 | = u \implies c x_1 + d x_2 = \pm u$ (and similarly for the $2nd$). Without further restrictions on $x_1$ and $x_2$, we can get valid solutions for $x_1$ and $x_2$ by choosing any one case from each of the $2$ sets of equations.