Converting a primal LP to a dual LP with a constant in the question

Question

Could someone help me with the next Primal LP to Dual LP conversion?

$$ min z = -3x_1 + x_2 - 20 \\ s.t. \quad -3x_1 + 3x_\le 6 \\ \quad\quad\qquad -8x_1 + 4x_2 \le 4 \\ \qquad\qquad x_1,x_2 \ge 0 $$

These problems usually aren't hard at all, but I'm unable to find what to do with the $-20$ in the top rule. What do I do with it in the Dual LP?

Thanks!

Answer 1

The set of $x$ that minimizes the objective function $f(x)$, minimizes also $f(x)+c$.

So in your case instead of $\min z = -3x_1 + x_2 - 20$ you can solve the problem using $\min z = -3x_1 + x_2 $. $$ \min z = -3x_1 + x_2 \\ s.t. \quad -3x_1 + 3x_2 \le 6 \\ \quad\quad\qquad -8x_1 + 4x_2 \le 4 \\ \qquad\qquad x_1,x_2 \ge 0 $$