How the Simplex method can be applied in solving the following system of Linear equations

Question

How the Simplex method can be applied in solving the following system of Linear equations? \begin{align*} x_1–x_3+4x_4 &=3 \\ 2x_1–x_2&=3 \\ 3x_1–2x_2–x_4&=1 \end{align*} Where $$ x_1,\;x_2,\;x_3,\;x_4\geq0 $$ What would be the proposed Objective function in this case? Further compare its solution to the solutions being obtained by other usual methods (Gaussian Elimination, Gauss-Jordan, etc…)

Answer 1

You can make up any linear objective function you like, e.g. $x_1$. The Simplex Method will find an optimal solution (which you don't care about), but more importantly, it will be a feasible solution, i.e. it will satisfy the equations.

The "diophantine equations" tag seems misplaced, since you don't require an integer solution, at least if I understand the problem correctly.