There are a set of equations like
$A_x + A_y + A_z = P$
$B_x + B_y + B_z = Q$
$C_x + C_y + C_z = R$
Where the values of only $P, Q, R$ are known.
Also, we have
$A_x + B_x + C_x = I$
$A_y + B_y + C_y = J$
$A_z + B_z + C_z = K$
where only the values of $I, J$ and $K$ are known.
Is there any way we know the individual values of
$A_x, B_x, C_x, A_y, A_z$ and the rest?
Substituting the above equations yield the result that $I + J + K = P + Q + R$ but how can I get the individual component values? Is any other information required to solve these equations?
Here's a good complementary question. If solutions exist, how to generate all of them? Are there some algorithms?