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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?

$$a=\frac{b.J_x - M_x}{J_{xy}}$$ $$b=\frac{M_y J_{xy}}{J_x J_y - M_x J_y + bJ^2_{xy}}$$

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Denote the equations by $(1)$ and $(2)$ respectively. Then

$$ J_x \times (2) - J_{xy} \times (1) \implies a(J_xJ_y-J_{xy}^2) = M_yJ_x+M_xJ_{xy} \implies a=\dfrac{M_yJ_x+M_xJ_{xy}}{J_xJ_y-J_{xy}^2}$$

$$ J_{xy}\times(2) - J_y\times (1) \implies b(J_{xy}^2 - J_yJ_x) = M_yJ_{xy} + M_xJ_y \implies b=-\dfrac{ M_xJ_y+ M_yJ_{yx} }{J_yJ_x-J_{yx}^2 }$$

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