A matrix with given row and column sums 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?

You are trying to solve
$$\left(\begin{matrix}
1&1&1&0&0&0&0&0&0\\
0&0&0&1&1&1&0&0&0\\
0&0&0&0&0&0&1&1&1\\
1&0&0&1&0&0&1&0&0\\
0&1&0&0&1&0&0&1&0\\
0&0&1&0&0&1&0&0&1
\end{matrix}\right) \left(\begin{matrix} A_x\\A_y\\A_z\\B_x\\B_y\\B_z\\C_x\\C_y\\C_z
\end{matrix}\right) = \left(\begin{matrix} P\\Q\\R\\I\\J\\K \end{matrix}\right)$$
If you have any solution, it will be at least a four*-dimensional solution space, and to have a solution, necessarily the equation $P+Q+R=I+J+K$ must hold, as you already found out.
*If the LHS is $Ax$ then the dimension (if $\ge 0$) is at least $9 - {\rm rg}(A) = 9-5 = 4$, the rank is only five because the sum of the first three rows is equal to the sum of the last three rows.
