I have a question concerning the formulation of a linear programmign task. I am trying fo find $x^* \in argmax_{x \in R^n}\{ a_1x_1 + a_2x_2, a_2x_2 + a_3x_3 + a_4x_a, a_4x_4 + a_5x_5 \}$, subject to $\sum_{i=1}^5 x_i = 3000, x_i \geq 0$, where $a_i$ are some coefficients $\in R$.
This can be formulated as a linear programming task like this:
\begin{equation} (x^*, \lambda^*) \in argmin_{x, \lambda} -\lambda \end{equation} Subject to:
\begin{equation} \begin{aligned} a_1x_1 + a_2x_2 \geq \lambda \\ a_2x_2 + a_3x_3 + a_4x_4 \geq \lambda \\ a_4x_4 + a_5x_5 \geq \lambda \\ \sum_{i=1}^5 x_i = 3000 \\ x_i \geq 0 \end{aligned} \end{equation}
I intuitively know, why I formulate it like this and that lambda is the "worst case" scenario (minimum value of my criterial function). What I would like to know is how exatly formally write why is the LP formulation equivalent to the first formulation I provided (what did I do to the first equation to get the LP one). Thanks in advance!