Recitation 1

Problem 1: Solve the system of equations:

\begin{aligned}x_1 + 2x_2 - x_3 &= 6 \\ 3x_1 + 8x_2 + 9x_3 &= 10 \\ 2x_1 - x_2 + 2x_3 &= -2 \end{aligned}

Comment: Use Gaussian elimination on the associated augmented matrix.

Problem 2: Solve graphically: maximize $z = 3x_1 + x_2$ subject to $2x_1 + x_3 \le 6, x_1 + 3x_2 \le 9, x_1 \ge 0, x_2 \ge 0$.

Solution: The red lines are related to the first two constraints. The reddish area is the feasible domain. The dashed blue lines are level lines on which $z=3,6,9$ respectively. From the graph, $z=9$ is maximized by $x_1 = 3, x_2 = 0$.

Problem 3: Solve graphically (should have non-unique solutions): maximize $z=4x_1 + x_2$ subject to $8x_1 + 2x_2 \le 16, 5x_1 + 2x_2 \le 12, x_1 \ge 0, x_2 \ge 0$.

Solution: The red lines are related to the first two constraints. The reddish area is the feasible domain. The dashed blue lines are level lines on which $z=4,8$ respectively. From the graph, $z=8$ is maximized by points between $A$ and $B$.

Solution: We want to maximize the total value $60x_1 + 70x_2 + 80x_3 + 90x_4 + 50x_5$, where $x_i$ is a variable that can assume only a 1 or a 0 depending on whether or not the item is on the probe. Those variables are subject to the total payload of 140 pounds, that is, $35x_1 + 45x_2 + 55x_3 + 42x_4 + 35x_5 \le 140$.