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Regions and linear programming
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A region on a graph shows every point that satisfies a set of inequalities. Linear programming finds the best point in that region.
Drawing the boundary
Draw the line as if it were an equation.
Use a solid line for ≤ or ≥ (the line is included).
Use a dashed line for < or > (the line is not included).
Use a solid line for ≤ or ≥ (the line is included).
Use a dashed line for < or > (the line is not included).
Choosing the side
Test a point not on the line, such as (0, 0).
For y < 2x + 1: 0 < 1 is true, so the side containing (0, 0) is wanted.
Shade the wanted or unwanted side, and say which you have done.
For y < 2x + 1: 0 < 1 is true, so the side containing (0, 0) is wanted.
Shade the wanted or unwanted side, and say which you have done.
Linear programming
To find the greatest or least value of an expression like P = x + y in a region, test it at each vertex (corner) of the region.
Region x ≥ 0, y ≥ 0, x + 2y ≤ 8, 3x + y ≤ 9 has vertices (0, 0), (3, 0), (0, 4) and (2, 3).
Region x ≥ 0, y ≥ 0, x + 2y ≤ 8, 3x + y ≤ 9 has vertices (0, 0), (3, 0), (0, 4) and (2, 3).
Find the maximum of P = 2x + 3y in the region x ≥ 0, y ≥ 0, x + 2y ≤ 8, 3x + y ≤ 9.
- Vertices: (0, 0), (3, 0), (0, 4), (2, 3)
- P = 0, 6, 12, 13
Answer: Maximum P = 13 at (2, 3)
Solid line for ≤ or ≥, dashed for < or >. Test a point to pick the side. Maximum and minimum values occur at vertices of the region.
The interactive lesson includes the diagrams for this topic.
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