First, we'll solve a linear programming problem with Sage. Although it is limited to solving problems in which the objective function and the constraints are linear functions of the variables, linear programming is widely taught in applied mathematics courses and has many practical applications. Let's see how Sage can help us visualize what's going on in an example problem from the Sage documentation. The problem to be solved is as follows:

Minimize –4x1–5x2 subject to the following linear inequality constraints:

2x1+x2≤3

x1+2x2≤3

x1≥0

x2≥0

var('x, y')
c=vector(RDF,[-4, -5])
G=matrix(RDF,[[2, 1], [1, 2], [-1, 0], [0, -1]])
h=vector(RDF,[3, 3, 0, 0])


sol=linear_program(c, G, h)
print("Minimum: {0}".format(sol['x']))
print("Slack variables: {0}" .format(sol['s']))
c1_plot = implicit_plot(2 * x + y == 3, (x,0,2), (y,0,2))
c2_plot = implicit_plot(x + 2 * y == 3, (x,0,2), (y,0,2))
c3_plot = implicit_plot(x == 0, (x,0,2), (y,0,2))
c4_plot = implicit_plot(y ==...