$$\text{Max } x_1 +3x_2$$ $$\text{s.t } 3x_1+x_2\leq 3$$ $$x_1-x_2\geq 2$$ $$x_1,..,x_2\geq 0$$
So we first transform to the standard form:
$$\text{Min } -x_1 -3x_2$$ $$\text{s.t } 3x_1+x_2+x_3= 3$$ $$x_1-x_2-x_4= 2$$ $$x_1,..,x_4\geq 0$$
Because there is no basic feasible solution we will add and artificial variable to move to a canonical form:
$$\text{Min } -x_1 -3x_2$$ $$\text{s.t } 3x_1+x_2+x_3= 3$$ $$x_1-x_2-x_4+x_5= 2$$ $$x_1,..,x_5\geq 0$$
Using Big M we will add $mx_5$ to the objective function:
$$\text{Min } -x_1 -3x_2+mx_5$$ $$\text{s.t } 3x_1+x_2+x_3= 3$$ $$x_1-x_2-x_4+x_5= 2$$ $$x_1,..,x_5\geq 0$$
But now, it is is not in basic form as the objective function need to be written in the non basic variables, so we will add -M times equation 2 to the objective function:
$$\text{Min } (-m-1)x_1 +(m-3)x_2+mx_4=-m-1$$ $$\text{s.t } 3x_1+x_2+x_3= 3$$ $$x_1-x_2-x_4+x_5= 2$$ $$x_1,..,x_5\geq 0$$
Now we can start the simplex:
moving $x_1$ to the basis we get:
But here all the coefficients of the objective function is positive as $m$ is a big positive number, how can we continue? or we can not and we still have $x_5$ is the basis so there is no feasible solution?