How to show that the set of all feasible solution of a linear programming problem is a convex set theoretically.
Let $x$ and $y$ lie in the feasible region and let $z = tx+(1-t)y$, where $t \in [0,1]$. We have to show that $z$ also lies in the feasible region. Since $x$ and $y$ are points in the feasible region, $x\geq 0, Ax \leq b$, and $y \geq 0, Ay \leq b$ by definition.
Now show that $z \geq 0$ and $Az \leq b$.