I have the following system: \begin{align} \left\{ \begin{array} $21 = 55x_{100} + 54x_{99} + \dots + x_{46} \\ 17 = 50x_{100} + 49x_{99} + \dots + x_{51} \\ 13 = 45x_{100} + 44x_{99} + \dots + x_{56} \\ 20 = 40x_{100} + 39x_{99} + \dots + x_{61} \\ \end{array} \right. \end{align} Does the system have a solution $x \in \mathbb{R}^{100}$ with all its elements strictly positive? I'm stuck in answering this question.
What is the quickest way to answer this question?
The question of whether such a solution exists can be formulated as a linear program.
We will introduce an auxiliary variable $t$, which at optimality will satisfy $t=\min\{x_1,x_2,\dots,x_{100}\}$. We wish to maximize $t$. If $t>0$, then $x_i>0$ for all $i=1,\dots,100$. Otherwise, no such solution exists.
The resulting linear program is $$ \begin{array}{rl} \max&t\\ \text{s.t.}&t\leq{x_i}\text{ for all }i=1,\dots,100\\ &t\leq1\\ &(\text{Your system of equalities here}) \end{array} $$ The constraint $t\leq1$ ensures that the problem isn’t unbounded.