Solutions of the LP problem

Question

I am asked to find all the solutions of the following linear programming problem using the simplex method.

$$\min(2x + 3y + 6z + 4w)$$ $$\begin{aligned} x+2y+3z+w &\geq 5\\ x+y+2z+3w &\geq 3\\ x,y,z,w &\geq 0 \end{aligned}$$

To solve this problem we use the Two-Phase method, don't we? Then I found that there are infinite many solutions, $(x,y,z,w)=(1-m, 2-0.4m, 0, 0.2m), 0≤m≤1$. Could you tell me if this is right?

Answer 1

$x+2y+3z+w \geq 5\\ x+y+2z+3w \geq 3 \\ \implies 2x+3y+5z+4w \ge 8 \implies 2x+3y+6z+4w \ge 8 +z \ge 8 $

so the min is $8$ when $z=0$