Convert the following linear program into standard form:

\[\begin{array}{lrcrcrcrl} \text{minimize} & 2x_1 & + & 7x_2 & + & x_3 & & \\ \text{subject to} & \\ & x_1 & & & - & x_3 & = & 7 \\ & 3x_1 & + & x_2 & & & \ge & 24 \\ & & x_2 & & & & \ge & 0 \\ & & & & & x_3 & \le & 0 & . \end{array}\]

To begin, we must convert this minimization linear program into an equivalent maximization linear program. This is as simple as negating the coefficients in the objective function:

\[\begin{array}{lrcrcrcrl} \text{maximize} & -2x_1 & - & 7x_2 & - & x_3 & & \\ \text{subject to} & \\ & x_1 & & & - & x_3 & = & 7 \\ & 3x_1 & + & x_2 & & & \ge & 24 \\ & & x_2 & & & & \ge & 0 \\ & & & & & x_3 & \le & 0 & . \end{array}\]

Next, we must attend to the fact that \(x_1\) and \(x_3\) do not have non-negativity constraints. For \(x_1\), we replace all instances of \(x_1\) with \(x'_1 - x''_1\) and for \(x_3\), we replace all instances of \(x_3\) with \(-x'_3\) in order to flip the inequality:

\[\begin{array}{lrcrcrcrl} \text{maximize} & -2x'_1 & + & 2x''_1 & - & 7x_2 & + & x'_3 & & \\ \text{subject to} & \\ & x'_1 & - & x''_1 & & & + & x'_3 & = & 7 \\ & 3x'_1 & - & 3x''_1 & + & x_2 & & & \ge & 24 \\ & x'_1 & , & x''_1 & , & x_2 & , & x'_3 & \ge & 0 & . \end{array}\]

Our first constraint is an equality, which is not valid in standard form. We can handle this by creating a pair of inequalities that express equality as so:

\[\begin{array}{lrcrcrcrl} \text{maximize} & -2x'_1 & + & 2x''_1 & - & 7x_2 & + & x'_3 & & \\ \text{subject to} & \\ & -x'_1 & + & x''_1 & & & - & x'_3 & \ge & 7 \\ & x'_1 & - & x''_1 & & & + & x'_3 & \le & 7 \\ & 3x'_1 & - & 3x''_1 & + & x_2 & & & \ge & 24 \\ & x'_1 & , & x''_1 & , & x_2 & , & x'_3 & \ge & 0 & . \end{array}\]

Every constraint that uses a greater-than-or-equal-to sign is not valid in standard form, so let’s flip them:

\[\begin{array}{lrcrcrcrl} \text{maximize} & -2x'_1 & + & 2x''_1 & - & 7x_2 & + & x'_3 & & \\ \text{subject to} & \\ & x'_1 & - & x''_1 & & & + & x'_3 & \le & -7 \\ & x'_1 & - & x''_1 & & & + & x'_3 & \le & 7 \\ & -3x'_1 & + & 3x''_1 & - & x_2 & & & \le & -24 \\ & x'_1 & , & x''_1 & , & x_2 & , & x'_3 & \ge & 0 & . \end{array}\]

Finally, let’s rename some variables. \(x'_1\) becomes \(x_1\), \(x''_1\) becomes the new \(x_2\), the old \(x_2\) becomes \(x_3\) and \(x'_3\) becomes \(x_4\):

\[\begin{array}{lrcrcrcrl} \text{maximize} & -2x_1 & + & 2x_2 & - & 7x_3 & + & x_4 & & \\ \text{subject to} & \\ & -x_1 & - & x_2 & & & + & x_4 & \le & -7 \\ & x_1 & - & x_2 & & & + & x_4 & \le & 7 \\ & -3x_1 & + & 3x_2 & - & x_3 & & & \le & -24 \\ & x_1 & , & x_2 & , & x_3 & , & x_4 & \ge & 0 & . \end{array}\]