SOLUTION: Minimize the expression Z= 2x +3y subject to the following constraints. y <= 5, x <=6, x+y >= 2, x >= 0, y >=0

Question 227686: Minimize the expression Z= 2x +3y subject to the following constraints.
y <= 5, x <=6, x+y >= 2, x >= 0, y >=0
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
To solve problems like this:

Graphs the constraints. This should form some form of polygon.
Find the coordinates of the vertices of the polygon.

Unfortunately Algebra.com's graphing software does not handle graphs like this very well. So I will have to describe the graph of the contraints. Each contraint will form a side of the polygon.

y <= 5: A horizontal line through 5 on the y-axis
x <= 6: a vertical line through 6 on the x-axis
x+y >= 2. In slope-intercept form this is y >= -x + 2. This is a line with a slope of -1 through 2 on the y-axis
x >=0: the y=axis
y >=0: the x-axis

Here's a partial graph (without the x <=6 line):

Remembering the missing vertical line through 6 on the x-axis, we end up with a pentagon (5-sided polygon). Any points that are within the pentagon or on the sides of the pentagon are possible solutions. The minimum (or maximum) value for Z will come from one of the vertices. So we need to find the vertices. By inspection (or with a little calculation) we can find the following vertices: (0, 2), (0, 5), (6, 5), (6, 0) and (2, 0).

Now all we need to do is use these vertices to find the minimum value for Z. Take each vertex, one at a time, and put its coordinates into the formula for Z. This will give you 5 different values for Z. The lowest one is the minimum.
I'll do one Z and leave the rest for you. The Z for the vertex (0, 2):
Z = 2(0) + 3(2) = 0 + 6 = 6
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