Convert the following linear program into slack form:

\[\begin{array}{lrcrcrcrl} \text{maximize} & 2x_1 & & & - & 6x_3 & & \\ \text{subject to} & \\ & x_1 & + & x_2 & - & x_3 & \le & 7 \\ & 3x_1 & - & x_2 & & & \ge & 8 \\ & -x_1 & + & 2x_2 & + & 2x_3 & \ge & 0 \\ & x_1 & , & x_2 & , & x_3 & \ge & 0 & . \end{array}\]

What are the basic and nonbasic variables?

To begin, let’s quickly convert this linear program into standard form by flipping the 2nd and 3rd constraints to be inequalities that use less-than-or-equal-to signs:

\[\begin{array}{lrcrcrcrl} \text{maximize} & 2x_1 & & & - & 6x_3 & & \\ \text{subject to} & \\ & x_1 & + & x_2 & - & x_3 & \le & 7 \\ & -3x_1 & + & x_2 & & & \le & -8 \\ & x_1 & - & 2x_2 & - & 2x_3 & \le & 0 \\ & x_1 & , & x_2 & , & x_3 & \ge & 0 & . \end{array}\]

Next, we introduce three new slack variables \(x_4\), \(x_5\), \(x_6\), obtaining:

\[\begin{array}{lrcrcrcrl} \text{maximize} & & & & & 2x_1 & & & - & 6x_3 & & \\ \text{subject to} & \\ & x_4 & = & 7 & - & x_1 & - & x_2 & + & x_3 \\ & x_5 & = & -8 & + & 3x_1 & - & x_2 \\ & x_6 & = & 0 & - & x_1 & + & 2x_2 & + & 2x_3 \\ & x_1, x_2, x_3, x_4, x_5, x_6 & \ge & 0 . \end{array}\]