if $J_x= \oint y^2 ds $ and
$J_y= \oint x^2 ds $ and
$J_{xy} = \oint xy ds $
how I can find $a$ and $b$?
$$\left\{\begin{matrix}
a.J_{xy}+b.J_x=-M_x\\
a.J_y+bJ_{xy}=M_y
\end{matrix}\right.$$
$M_x, M_y$ are const.
Is it a right solution to find $a$ based on $b$ from one of the equations and replace its value on the other equation?
$$a=\frac{b.J_x - M_x}{J_{xy}}$$ $$b=\frac{M_y J_{xy}}{J_x J_y - M_x J_y + bJ^2_{xy}}$$
Denote the equations by $(1)$ and $(2)$ respectively. Then
$$ J_x \times (2) - J_{xy} \times (1) \implies a(J_xJ_y-J_{xy}^2) = M_yJ_x+M_xJ_{xy} \implies a=\dfrac{M_yJ_x+M_xJ_{xy}}{J_xJ_y-J_{xy}^2}$$
$$ J_{xy}\times(2) - J_y\times (1) \implies b(J_{xy}^2 - J_yJ_x) = M_yJ_{xy} + M_xJ_y \implies b=-\dfrac{ M_xJ_y+ M_yJ_{yx} }{J_yJ_x-J_{yx}^2 }$$