System of Inequalities - Linear Programming
System of Inequalities - How it Works - Video
Guidelines
Guidelines:
Here we have the guidelines to solve for any linear programming problem.
Example 1
Example 1:
We have a system of inequalities. It is just like a system of equations but we have to shade.
We have four inequalities x - 2y ≤ -12, 2x + y ≤ 16, x ≥ 0, and y ≥ 0. After graphing and shading, we get an area in Q 1. The shaded represents all the possible values that one could pick to run operations.
Each vertex has a possibility to be a maximum or a minimum depending on the equation. We have four possibilities: (0, 0), (0, 6), (4, 8), and (8, 0).
(0, 0) => 2 * 0 + 3 * 0 => 0 + 0 => 0
(0, 0) => 2 * 0 + 3 * 6 => 0 + 18 => 18
(0, 0) => 2 * 4 + 3 * 8 => 8 + 24 => 32
(0, 0) => 2 * 8 + 3 * 0 => 16 + 0 => 16
The vertex (4, 8) gives us the best case scenario or the highest profit.
To test if we did it correctly, we solve for y and graph a line with the same slope through each vertex.
Our slope is -2/3 and we have drawn a line through each vertex. To find the maximum or minimum we check all the y-intercepts. What we have drawn here matches the minimum and maximum values we found substituting the vertices into the function.