\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)
Simplex method
Tableau, pivot step, pivot rules. Exceptions. Phase one (finding a feasible point).
Sample problem

For which \(h\) the following system is feasible?
\[
\begin{array}{ccc}
&y&\leq 0\\
x&&\geq 0\\
x&y&\leq 4h\\
x&+y&\leq 2h\\
x&+y&\leq h+2\\
\end{array}
\] 
Using simplex method, solve
\[
\begin{array}{ccl}
2x&3y&\to\min, \mathrm{subj\ \ to}\\
3x&+2y&\leq 6\\
x&+2y&\leq 10\\
x&+y&\leq 8\\
3x&+y&\leq 18\\
x&y&\leq 2\\
x&&\geq 0\\
&y&\geq 0\\
\end{array}
\]
Does question 2 also have x,y >= 0? Or we don’t know.
No, it is not assumed.