Here's how you can do it in Mathematica:

Define the problem and variables: Let x be the number of 19" pieces to cut from each 2x4, and y be the number of 7" pieces to cut from each 2x4.

Define the objective function: The objective is to minimize the waste, which can be represented as (96 - (19x + 7y)), as 96 is the length of each 2x4 and (19x + 7y) is the total length of pieces cut from it.

Set up the constraints: You have the constraint that the total length of pieces cut from each 2x4 cannot exceed 96 inches, so the constraint is (19x + 7y <= 96).

Use Mathematica to solve the linear programming problem:

(* Load the LinearProgramming package *) Needs["LinearProgramming`"]

(* Define the objective function coefficients and constraints *) objectiveCoefficients = {19, 7}; constraintsMatrix = {{19, 7}}; constraintsVector = {96};

(* Set up the optimization problem *) solution = LinearProgramming[objectiveCoefficients, constraintsMatrix, constraintsVector];

(* Extract the values of x and y from the solution *) {x, y} = solution;

(* Calculate the waste *) waste = 96 - (19x + 7y);

(* Print the results *) Print["Number of 19\" pieces (x) per 2x4: ", x]; Print["Number of 7\" pieces (y) per 2x4: ", y]; Print["Waste: ", waste, " inches"];

I forgot I had chosen 19" because 96"/5 already minimizes the waste in cutting a 2x4 of that length.

But I'm still curious how Mathematica would solve such an optimization problem.

Here is one way:

model = 96 - 19 x - 7 y;
sols0 = List @@ Reduce[model >= 0, {x, y}, NonNegativeIntegers];
sols = SortBy[{#, Simplify[model, #]} & /@ sols0, Last];
(* result: *)
First[sols]
(*  Out{x\[Equal]1&&y\[Equal]11,0}   *)