I have a set of equations as follow,
- $a_1\cdot x_1 + a_2\cdot x_2 + a_3\cdot x_3 + a_4\cdot x_4 + \dots + a_{11} \cdot x_{12} + a_{12}\cdot x_{12} = 6000$
- $x_1 + x_2 + x_3 + x_4 + \dots x_{11} + x_{12} = 100 $
- $\{a_1,a_2,a_3, \dots , a_{11}, a_{12}\} \leq 60$
How can I find the values for all $a_i, \text{ and } x_i$,not all values of $a_i$ can be equal to $60$ also if possible integer, else the general solution.
- How the question get affected if I change the constrain on $a_i$ to some $X$, say $0 < a_i \leq X$,
- Also if the constrain on $a_i$ is , say $a_i>0$.
If solvable in Matlab please show the code also .
If not solvable please mention why.
What I tried is $$ \begin{bmatrix} a_1&a_2 & a_3 &\dots & a_{11} & a_{12} \\ 1 & 1 & 1& \dots & 1 & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots\\ x_{11} \\ x_{12} \end{bmatrix} = \begin{bmatrix} 6000 \\ 100 \end{bmatrix} $$
The rank will be almost $2$ so atmost I can get $10$ free variables. but since the values for $a_i$'s are also unknown I don't know how to proceed.