Linear programming and the simplex method

Question

I am trying to solve this system of equations. My approach would be to introduce slack variables and then somehow use the simplex algorithm to solve this. Can anyone show me how this is done?

Answer 1

For the simplex method indeed consult your book or online solvers. I understand that my answer does not exactly answer your question but in gives you an alternative way to solve it or to check that your result is correct.

For that end consider the dual linear program $$\min 5y_1+10y_2$$ subject to the constraints $$\begin{align*}&y_1+6y_2\ge 36 \\& y_1+5y_2\ge 30\\-&y_1+0y_2\ge -3 \\0&y_1-1y_2\ge -4 \\&\phantom{-1}y_1,y_2\ge0 \end{align*}$$ Constraints (3) and (4) imply that $$y_1\le 3, \qquad \text{and }\qquad y_2\le 4$$ but then subsituting in the first inequality you have that $$y_1+6y_2\le3+6\cdot4=27\not\ge36$$ Thus the feasible region of the dual is empty which implies that the primary problem is unbounded.