Having the following:
\begin{equation*} \begin{cases} \max& 3 x_1 & + 2x_2 & +4x_3\\ &x_1 &+ x_2 &+ 2 x_3 &\le 4\\ &2x_1 & &+3 x_3 &\le7\\ &2 x_1 &+ x_2 &+x_3 &\le 7\\ &\forall i,x_i\ge 0 \end{cases} \end{equation*} resulting, after one iteration, in the following Simplex programm: $$ \begin{array}{llllll} x_3= \frac{5}{3}& -\frac{2}{3}X_1 & & & &-\frac{1}{3}X_5\\ x_4= \frac{2}{3}& +\frac{1}{3} X_1& -X_2& &\\ x_6= 2& & -X_2 & & &+X_5 \\ \hline z= \frac{20}{3} &+\frac{1}{3}X_1 & + 2X_2 & & & -\frac{4}{5}X_5 \end{array} $$
Can $X_1$ enter the basis and $X_4$ exit it?
We would have:
$$X_1=-2 + 3 X_2 + 3 X_4 -2 X_5$$
Having such a $-2$ leaves me perplex, shouldn't any $X_i$ be positive?
Thus what are the conditions to have an operation be lawful in the Simplex method?