The problem that i tried to solve have this particular case:
Problem : In $R^3$, Let $P_1$ be the plane $x+y+z=l$, $l$ greater than $0$, and let $P_2$ be a generic plane with equation $a x+b y+c z=0$, with {a,b,c} not all with the same sign (otherwise, our problem would'n be well definied since the vector (a,b,c) would be in the first octant). Find the 2 volumes of the regions determined by the the 2 planes and the first octant.
The general case is a generalization of this problem to higher dimensions that i come up and can't solve it: In $R^n$, Let $P_1$ be the plane $x_1+x_2+ ... + x_n=l$, $l$ greater than $0$, and let $P_2$ be a generic plane with equation $a_1 x_1+a_2 x_2+...+a_nx_n=0$. Supposing that this problem is well defined, Find the 2 volumes of the regions determined by the the 2 planes and the first "octant" (i.e., the points $(x_1,x_2,...,x_n)$ with $x_i>0, i = 1,...,n$).
Let us consider the two planes with equations:
$$\begin{cases}(P_1): \ &x+y+z&=&L\\(P_2): \ &ax+by&=&z \end{cases}$$
(with an unimportant difference of notation with you for the second equation),
The intersection of plane $(P_1)$ with the first octant is an equilateral triangle (yellowish). Here we have taken $L=3$. Plane $(P_2)$ (dark green) has here equation $z=-2x+3y.$
we obtain intersection points with:
$$\begin{cases}A(0,L/(b+1),Lb/(b+1))\\B(-bL/(a-b),aL/(a-b),0)\\C(0,L,0)\end{cases}$$
The volume of tetrahedron $OABC$ will be obtained classically as
$$\dfrac16 \det[\vec{OA},\vec{OB},\vec{OC}]=\dfrac{L^3b^2}{6(b - a)(b + 1)}$$
The other volume can be easily computed by difference.