The image shows a linear programming problem: Objective Function: \[ Z = 40x_1 + 30x_2 \] Subject to the constraints: \[ \begin{align*} x_1 + x_2 & \leq 12 \\ 2x_1 + x_2 & \leq 16 \\ x_1 & \geq 0 \\ x_2 & \geq 0 \\ \end{align*} \] Additionally, there's a note discussing slack variables, indicating that they are used to convert inequalities into equalities within linear programming.

Graph the following system {-3x+y<1  -x+y<=2

Name the three types of solutions linear systems can have and give a graphical example of each type using two variables.

Suppose the system 5x3 7x4 - x2 x2 | 2x2 + + 16x4 x3 has infinitely many solutions. Can you say anything about the number of solutions to the following system? 5x3 7x4 10 x2 x2 x3 2x2 + 16x4 20 x3 + +