Linear Programming

Any linear programming problem will have some fixed restrictions that create inequalities. 

These can be used to draw a region in the Cartesian plane to reduce the number of possibilities to compare to find the greatest or least value.

Shading the unwanted region. In this diagram, the inequalities x + 3y ⩽ 4,      x – 4y ⩽ 4 and y – 2x ⩽ 6 

The possible values for consideration are then left unshaded.

Hint: You may use geogebra.org  to help you drawing the inequalities

To find the greatest or least value for an expression it is not necessary to work out every value each time.

The expression of 2x+y has values shown on the grid.

When a line is drawn through the zero values it creates a slider that can be used to find the greatest and least values.

The greatest or least values within a particular region can be found by using the line over the region.

As the line moves across the diagram the last point within the region it touches will be the greatest value.

In most cases, the last point that the line touches will be one of the corners.

You can evaluate the expression at each corner point to find which one is the greatest or least value.

Find the greatest and least values of 2x+y

Check the values of 2x+y using the three corners (4,0), (-2,2) and (-4,-2)

For (4,0) --> 2x+y --> 2.4+0=8